Finitary, Causal and Quantal Vacuum Einstein Gravity

Abstract We continue recent work [74, 75] and formulate the gravitational vacuum Einstein equations over a locally ?nite spacetime by using the basic axiomatics, techniques, ideas and working philosophy of Abstract Di?erential Geometry. The main kinematical structure involved, originally introduced and explored in [74], is a curved principal ?nitary spacetime sheaf of incidence algebras, which have been interpreted as quantum causal sets, together with a non-trivial locally ?nite spin-Loretzian connection on it which lays the structural foundation for the formulation of a covariant dynamics of quantum causality in terms of sheaf morphisms. Our scheme is innately algebraic and it supports a categorical version of the principle of general covariance that is manifestly independent of a background C ∞ -smooth spacetime manifold M . Thus, we entertain the possibility of developing a ‘fully covariant’ path integral-type of quantum dynamical scenario for these connections that avoids ab initio various problems that such a dynamics encounters in other current quantization schemes for gravity—either canonical (Hamiltonian), or covariant (Lagrangian)—involving an external, base di?erential spacetime manifold, namely, the choice of a di?eomorphism-invariant measure on the moduli space of gauge-equivalent (self-dual) gravitational spin-Lorentzian connections and the (Hilbert space) inner product that could in principle be constructed relative to that measure in the quantum theory—the so-called ‘inner product problem’, as well as the ‘problem of time’ that also involves the Di?(M ) ‘structure group’ of the classical C ∞ -smooth spacetime continuum of general relativity. Hence, by using the inherently algebraico-sheaf-theoretic and calculus-free ideas of Abstract Di?erential Geometry, we are able to draw preliminary, albeit suggestive, connections between certain non-perturbative (canonical or covariant) approaches to quantum general relativity (eg, Ashtekar’s new variables and the loop formalism that has been developed along with them) and Sorkin et al.’s causal set program—as it were, we ‘noncommutatively algebraize’, ‘di?erential geometrize’ and, as a result, dynamically vary causal sets. At the end, we anticipate various consequences that such a scenario for a locally ?nite, causal and quantal vacuum Einstein gravity might have for the obstinate from the viewpoint of the smooth continuum problem of C ∞ smooth spacetime singularities, thus we prepare the ground for a forthcoming paper [76].PACS numbers: 04.60.-m, 04.20.Gz, 04.20.-q Key words: quantum gravity, causal sets, di?erential incidence algebras of locally ?nite partially ordered sets, abstract di?erential geometry, sheaf theory, sheaf cohomology, category theory Algebra and Geometry Section, Department of Mathematics, University of Athens, Panepistimioupolis Zografou 157 84, Athens, Greece; e-mail: amallios@cc.uoa.gr ? Theoretical Physics Group, Blackett Laboratory, Imperial College of Science, Technology and Medicine, Prince Consort Road, South Kensington, London SW7 2BZ, UK; e-mail: i.raptis@ic.ac.uk?

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Anastasios Mallios and Ioannis Raptis

“...the theory that space is continuous is wrong, because we get...in?nities [viz. ‘singularities’] and other similar di?culties ...[while] the simple ideas of geometry, extended down to in?nitely small, are wrong...” [39]

.................................. “...at the Planck-length scale, classical di?erential geometry is simply incompatible withquantum theory...[so that] one will not be able to use di?erential geometry in the true quantum-gravity theory...” [52]

1

Prologue cum Physical Motivation

In the last century, the path that we have followed to unite quantum mechanics with general relativity into a coherent, both technically and conceptually, quantum theory of gravity has been a long and arduous one, full of unexpected twists and turns, surprising detours, branchings and loops—even disheartening setbacks and impasses, as well as hopes, disappointments or even disillusionments at times. Certainly though, the whole enterprize has been supported and nurtured by impressive technical ingenuity, and creative imagination coming from physicists and mathematicians alike. All in all, it has been a trip of adventure, discovery and intellectual reward for all who have been privileged to be involved in this formidable quest. Arguably then, the attempt to arrive at a conceptually sound and ‘calculationally’ ?nite quantum gravity must be regarded and hailed as one of the most challenging and inspired endeavors in theoretical physics research that must be carried over and be zestfully continued in the new millenium. Admittedly, however, a cogent theoretical scenario for quantum gravity has proved to be stubbornly elusive not least because there is no unanimous agreement about what ought to qualify as the ‘proper’ approach to a quantum theoresis of spacetime and gravity. Generally speaking, most of the approaches fall into the following three categories:1 1. ‘General relativity conservative’: The general aim of the approaches falling into this category is to quantize classical gravity somehow. Thus, the mathematical theory on which general relativity—in fact, any ?eld theory whether classical or quantum—is based, namely, the di?erential geometry of C ∞ -manifolds (ie, the usual di?erential calculus on manifolds), is essentially retained2 and it is used to treat the gravitational ?eld quantum ?eld-theoretically. Both the non-perturbative canonical and covariant (ie, path integral or ‘action-weighed sum-over-histories’) approaches to ‘quantum general relativity’, topological quantum ?eld theories, as well as, to a large extent, higher dimensional (or extended objects’) theories like (super)string and membrane schemes arguably belong to this category.These categories should by no means be regarded as being mutually exclusive or exhaustive, and they certainly re?ect only these authors’ subjective criteria and personal perspective on the general characteristics of various approaches to quantum gravity. This coarse classi?cation will be useful for the informal description of our ?nitary and causal approach to Lorentzian vacuum quantum gravity to be discussed shortly. 2 That is, in general relativity spacetime is modelled after a C ∞ -smooth manifold. Purely mathematically speaking, approaches in this category could also be called ‘C ∞ -smoothness or di?erential manifold conservative’.1

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2. ‘Quantum mechanics conservative’: The general spirit here is to start from general quantum principles such as algebraic operationality, noncommutativity and ?nitism (‘discreteness’) about the structure of spacetime and its dynamics, and then try to derive somehow general relativistic attributes, as it were, from within the quantum framework. Such approaches assume up-front that quantum theory is primary and fundamental, while the classical geometrical smooth spacetime continuum and its dynamics secondary and derivative (emergent) from the deeper quantum dynamical realm. For instance, Connes’ noncommutative geometry [24, 57] and, perhaps more notably, Finkelstein’s quantum relativity [45, 106]3 may be classi?ed here. 3. ‘Independent’: Approaches in this category assume neither quantum mechanics nor general relativity as a fundamental, ‘?xed’ background theory relative to which the other must be modi?ed to suit. Rather, they start independently from principles that are neither quantum mechanical nor general relativistic per se, and proceed to construct a theory and a suitable mathematical formalism to accompany it that later may be interpreted as a coherent amalgamation (or perhaps even extension) of both. It is inevitable with such ‘iconoclastic’ schemes that in the end both general relativity and quantum mechanics may appear to be modi?ed to some extent. One could assign to this category Penrose’s combinatorial spin-networks [83, 101] and its current relativistic spin-foam descendants [13, 15, 84], Regge’s homological spacetime triangulations and simplicial gravity [94], as well as Sorkin et al.’s causal sets [18, 115, 116, 95, 117]. It goes without saying that this is no place for us to review in any detail the approaches mentioned above.4 Rather, we wish to continue a ?nitary, causal and quantal sheaf-theoretic approach to spacetime and vacuum Lorentzian gravity that we have already started to develop in [74, 75]. This approach, as we will argue subsequently, combines characteristics from all three categories above and, in particular, the mathematical backbone which supports it, Abstract Di?erential Geometry (ADG) [67, 68, 70, 71, 73], was originally conceived in order to evade the C ∞ -smooth spacetime manifold M (and consequently its di?eomorphism group Di?(M )) underlying (and creating numerous problems for) the various approaches in 1. For, it must be emphasized upfront, ADG is an axiomatic formulation of di?erential geometry which does not use any C ∞ -notion from the usual di?erential calculus—the classical di?erential geometry of smooth manifolds. To summarize brie?y what we have already accomplished in this direction,5 in [74] we comIn fact, Finkelstein maintains that “all is quantum. Anything that appears to be classical has not yet been resolved into its quantum elements” (David Finkelstein in private communication). 4 For reviews of and di?erent perspectives on the main approaches to quantum gravity, the reader is referred to [54, 4, 99]. In the last, most recent reference, one notices a similar partition of the various approaches to quantum gravity into three classes called covariant, canonical and sum-over-histories. Then one realizes that presently we assigned all these three classes to category 1, since our general classi?cation criterion is which approaches, like general relativity, more-or-less preserve a C ∞ -smooth base spacetime manifold hence use the methods of the usual di?erential geometry on it, and which do not. Also, by ‘covariant’ we do not mean what Rovelli does. ‘Covariant’ for us is synonymous to ‘action-weighed sum-over-histories’ or ‘path integral’. Undoubtedly, there is arbitrariness and subjectivity in such denominations, so that the boundaries of those distinctions are rather fuzzy. 5 For a recent, concise review of our work so far on this sheaf-theoretic approach to discrete Lorentzian quantum3

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bined ideas from the second author’s work on ?nitary spacetime sheaves6 (?nsheaves) [87] and on an algebraic quantization scenario for Sorkin’s causal sets (causets) [86] with the ?rst author’s ADG [67, 68], and we arrived at a locally ?nite, causal and quantal version of the kinematical structure of Lorentzian gravity. The latter pertains to the de?nition of a curved principal ?nsheaf Pi↑ of incidence Rota algebras modelling quantum causal sets (qausets) [86], having for structure group a locally ?nite version of the continuous orthochronous Lorentz group SO (1, 3)↑ of local symmetries (isometries) of general relativity, together with a non-trivial (ie, non-?at) locally ?7 nite so(1, 3)↑ i ? sl(2, C)i -valued spin-Lorentzian connection Di which represents the localization or gauging and concomitant dynamical variability of the qausets in the sheaf due to a ?nitary, causal and quantal version of Lorentzian gravity in the absence of matter (ie, vacuum Einstein gravity). We also gave the following quantum particle interpretation to this reticular scheme: a so-called causon—the elementary particle of the ?eld of dynamical quantum causality represented by Ai —was envisioned to dynamically propagate in the reticular curved spacetime vacuum represented by the ?nsheaf of qausets under the in?uence of ?nitary Lorentzian (vacuum) quantum gravity. In the sequel [75], by using the universal constructions and the powerful sheaf-cohomological tools of ADG together with the rich di?erential structure with which the incidence algebras modelling qausets are equipped [92, 86, 93, 130, 128], we showed how basic di?erential geometric ideas and results usually thought of as being vitally dependent on C ∞ -smooth manifolds for their ˇ realization, as for example the standard Cech-de Rham cohomology, carry through virtually unaltered to the ?nitary regime of the curved ?nsheaves of qausets. For instance, we gave ?nitary versions of important C ∞ -theorems such as de Rham’s, Weil’s integrality and the ChernWeil theorem and, based on certain robust results from the application of ADG to the theory of geometric (pre)quantization [68, 69, 72], we carried out a sheaf-cohomological classi?cation of the associated line sheaves bearing the ?nitary spin-Lorentzian Ai s whose quanta were referred to as causons above—the elementary (bosonic) particles carrying the dynamical ?eld of quantum causality whose (local) states correspond precisely to (local) sections of those line sheaves. By this virtually complete transcription of the basic C ∞ -constructions, concepts and results to the locally ?nite and quantal realm of the curved ?nsheaves of qausets, we highlighted that for their formulation the classical smooth background spacetime continuum is essentially of no contributing value. Moreover, we argued that since the C ∞ -smooth spacetime manifold can be regarded as the main culprit for the singularities that plague general relativity as well as for the weaker but still troublesome in?nities that assail the ?at quantum ?eld theories of matter, its evasion—especially by the ?nitistic-algebraic means that we employed—should be most welcome for the formulationgravity, as well as on its possible topos-theoretic extension, the reader is referred to [90]. In particular, the topostheoretic viewpoint is currently being elaborated in [91]. 6 Throughout this paper, the epithets ‘?nitary’ and ‘locally ?nite’ will be used interchangeably. 7 From [74] we note that only the gauge potential Ai part of the reticular Di = ?i + Ai is spin-Lorentzian proper (ie, discrete so(1, 3)↑ i ? sl (2, C)i -valued), but here too we will abuse terminology and refer to either Di or its part Ai as ‘the spin-Lorentzian connection’. (The reader should also note that the arrows over the various symbols will be justi?ed in the sequel in view of the causal interpretation that our incidence algebra ?nsheaves have; while, the subscript ‘i’ is the so-called ‘?nitarity’, ‘resolution’, or ‘localization index’ [87, 74, 75], which we will also explain in the sequel.)

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of a ‘calculationally’ and, in a sense to be explained later, ‘inherently ?nite’ and ‘fully covariant’ quantum theory of gravity. With respect to the aforementioned three categories of approaches to quantum gravity, our scheme certainly has attributes of 2 as it employs ?nite dimensional non-abelian incidence algebras to model (dynamically variable) qausets in the stalks of the relevant ?nsheaves, which qausets have a rather natural quantum-theoretic (because algebraico-operational) physical interpretation [92, 86, 74, 93]. It also has traits of category 3 since the incidence algebras are, by de?nition, of combinatorial and ‘directed simplicial’ homological character and, in particular, Sorkin’s causet theory was in e?ect its principal physical motivation [86, 75]. Finally, regarding category 1, the purely mathematical, ADG-based aspect of our approach was originally motivated by a need to show that all the ‘intrinsic’ di?erential mechanism of the usual calculus on manifolds is independent of C ∞ -smoothness, in fact, of any notion of ‘space’ supporting the usual di?erential geometric concepts and constructions,8 thus entirely avoid, or better, manage to integrate or ‘absorb’ into the (now generalized) abstract di?erential geometry, the ‘anomalies’ (ie, the singularities and other ‘in?nity-related pathologies’) that plague the classical C ∞ -smooth continuum case [67, 68, 73]. Arguably then, our approach is an amalgamation of elements from 1–3. Let us now move on to speci?cs. In the present paper we continue our work in [74, 75] and formulate the dynamical vacuum Einstein equations in Pi↑ . On the one hand, this extends our work on the kinematics of a ?nitary and causal scheme for Lorentzian quantum gravity developed in [74] as it provides a suitable dynamics for it, and on the other, it may be regarded as another concrete physical application of ADG to the locally ?nite, causal and quantum regime, and all this in spite of the C ∞ -smooth spacetime manifold, in accord with the spirit of [75]. Our work here is the second physical application of ADG to vacuum Einstein gravity, the ?rst having already involved the successful formulation of the vacuum Einstein equations over spaces with singularities concentrated on arbitrary closed nowhere dense sets—arguably, the most singular spaces when viewed from the featureless C ∞ -smooth spacetime manifold perspective [70, 77, 78, 73, 97]. The paper is organized as follows: in the following section we recall the basic ideas about connections in ADG focusing our attention mainly on Yang-Mills (Y-M) and Lorentzian connections on ?nite dimensional vector sheaves, on principal sheaves (whose associated sheaves are the aforementioned vector sheaves), their curvatures, symmetries and (Bianchi) identities, as well asThus, as we will time and again stress in the sequel, with the development of ADG we have come to realize that the main operative role of the C ∞ -smooth manifold is to provide us with a convenient (and quite successful in various applications to both classical and quantum physics!), but by no means unique, di?erential mechanism, namely, that accommodated by the algebra C ∞ (M ) of in?nitely di?erentiable functions ‘coordinatizing’ the (points of the) di?erential manifold M . However, the latter algebra’s pathologies in the form of singularities made us ponder on the question whether the di?erential mechanism itself is ‘innate’ to C ∞ (M ) and the manifold supporting these ‘generalized arithmetics’ (this term is borrowed straight from ADG). As alluded to above, ADG’s answer to the latter is an emphatic ‘No!’ [67, 68, 73]. For example, one can do di?erential geometry over very (in fact, the most!) singular from the point of view of the C ∞ -smooth M spaces and their ‘arithmetic algebras’, such as Rosinger’s non-linear distributions—the so-called di?erential algebras of generalized functions) [77, 78, 97]. As a matter of fact, the last two papers, together with the duet [74, 75], are examples of two successful applications of ADG proving its main point, that: “di?erentiability is independent of C ∞ -smoothness” (see slogan 2 at the end of [75]).8

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the a?ne spaces that they constitute. In section 3 we discuss the connection-based picture of gravity—the way in which general relativity may be thought of as a Y-M-type of gauge theory in the manner of ADG [71]. Based mainly on [70], we present vacuum Einstein gravity ` a la ADG and explore the relevant gravitational moduli spaces of spin-Lorentzian connections. In section 4 we remind the reader of some basic kinematical features of our curved principal ?nsheaves of qausets from [74, 75] and, in particular, based on recent results of Papatrianta?llou and Vassiliou [81, 82, 124, 125, 126], we describe in a categorical way inverse (projective) and direct (inductive) limits of such principal ?nsheaves and their reticular connections. We also comment on the use of the real (R) and complex (C) number ?elds in our manifold-free, combinatory-algebraic theory, and compare it with some recent critical remarks of Isham [55] about the a priori assumption—one that is essentially based on the classical manifold model of spacetime—of the R and C continua in conventional quantum theory vis-` a-vis its application to quantum gravity. Section 5 is the focal area of this paper as it presents a locally ?nite, causal and quantal version of the vacuum Einstein equations for Lorentzian gravity. The idea is also entertained of developing a possible covariant quantization scheme for ?nitary Lorentzian gravity involving a path integral-type of functional over the moduli space Ai /Gi of all reticular gauge-equivalent spin-Lorentzian connections Ai . Based on the ‘innate’ ?niteness of our model, we discuss how such a scenario may on the one hand avoid ab initio the choice of measure for Ai that troubles the continuum functional integrals over the in?nite dimensional, non-linear and with a ‘complicated’ topology moduli space (+) A∞ /G of smooth, (self-dual) Lorentzian connections in the standard covariant approach to the quantization of (self-dual) Lorentzian gravity, and on the other, how our up-front avoiding of Di?(M ) may cut the ‘Gordian knot’ that the problems of time and of the inner product in the Hilbert space of physical states present to the non-perturbative canonical approach to quantum gravity based on Ashtekar’s new variables and the holonomy (Wilson loop) formalism associated with them. Ultimately, all this points to the fact that our theory is genuinely C ∞ -smooth spacetime background independent and, perhaps more importantly, regardless of the perennial debate whether classical (vacuum) gravity should be quantized covariantly or canonically. This makes us ask—in fact, altogether doubt—whether quantizing classical spacetime and gravity by using the constructions and techniques of the usual di?erential geometry of smooth manifolds is the ‘right’ approach to quantum spacetime and gravity, thus align ourselves more with the categories 2 and 3 above, and less with 1. As a matter of fact, and in contradistinction to the ‘iconoclastic’ approaches in category 3 (most notably, in contrast to the theory of causal sets), in developing our entirely algebraico-sheaf-theoretic approach to ?nitary Lorentzian quantum gravity based on ADG, we have come to question altogether whether the notion of (an inert geometrical background) ‘spacetime’—whether it is modelled after a continuous or a discrete base space—makes any physical sense in the ever dynamically ?uctuating quantum deep where the vacuum is ‘?lled’ solely by (the dynamics of) causons and where there is no ‘ambient’ or surrounding spacetime that actively participates into or in?uences in any way that dynamics.9 We thus infer that bothOf course, we will see that there is a base topological ‘localization space’—a stage on which we solder our algebraic structures, but this space is of an ether-like character, a surrogate sca?olding of no physical signi?cance whatsoever as it does not actively engage into the quantum dynamics of the causons—the quanta of the ?eld Ai of quantum causality that is localized (gauged) and dynamically propagates on ‘it’.9

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our ?nitary vacuum Einstein equations for the causon and the path integral-like quantum dynam(+) ics of our reticular (self-dual) spin-Lorentzian connections Ai is ‘genuinely’, or better, ‘fully’ covariant since they both concern directly and solely the objects (the quanta of causality, ie, the dynamical connections Ai ) that live on that base ‘space(time)’, and not at all that external, passive and dynamically inert ‘space(time) arena’ itself. We also make comments on geometric (pre)quantization [68, 69, 72] in the light of our application here of ADG to ?nitary and causal Lorentzian gravity [75] and we stress that our scheme may be perceived as being, in a strong sense, ‘already’ or ‘inherently’ quantum, meaning that it is in no need of the (formal) process of quantization of the corresponding classical theory (here, general relativity on a C ∞ -smooth spacetime manifold). This seems to support further our doubts about the quantization of classical spacetime and gravity mentioned above. Furthermore, motivated by the ‘full covariance’ and ‘inherent quantumness’ of our theory, we draw numerous close parallels between our scenario and certain ideas of Einstein about the so-called (post general relativity) ‘new ether’ concept, the unitary ?eld theory that goes hand in hand with the latter, but more importantly, about the possible abandonement altogether, in the light of singularities and quantum discontinuities, of this continuous ?eld theory and the C ∞ -spacetime continuum supporting it for “a purely algebraic description of reality” [38]. In toto, we argue that ADG, especially in its ?nitary and causal application to Lorentzian quantum gravity in the present paper, may provide the basis for the “organic” [36], “algebraic” [38] theory that Einstein was searching for in order to replace the multiply assailed by unmanageable singularities, unphysical in?nities and other anomaliles geometric spacetime continuum of macroscopic physics. At the same time, we will maintain that this abandonement of the spacetime manifold for a more ?nitistic-algebraic theory can be captured to a great extent by the mathematical notion of Gel’fand duality—a notion that permeates the general sheaf-theoretic methods of ADG e?ectively ever since its inception [65, 66, 67] as well its particular ?nitary, causal and quantal applications thereafter [92, 86, 87, 74, 93, 75, 88, 89, 90]. The paper concludes with some remarks on C ∞ -smooth singularities—some of which having already been presented in a slightly di?erent, purely ADG-theoretic, guise in [73]—that anticipate a paper currently in preparation [76].

2

Connections in Abstract Di?erential Geometry

Connections, alias ‘generalized di?erentials’, are the central objects in ADG which purports to abstract from, thus axiomatize and e?ectively generalize, the usual di?erential calculus on C ∞ -manifolds. In this section we give a brief r? esum? e of both the local and global ADG-theoretic perspective on linear (Koszul), pseudo-Riemannian (Lorentzian) connections and their associated curvatures. For more details and completeness of exposition, the reader is referred to [67, 68, 71].

2.1

Basic De?nitions about Linear Connections

The main notion here is that of di?erential triad T = (AX , ?, ?(X )), which consists of a sheaf AX of (complex) abelian algebras A over an in general arbitrary topological space X called the

for any local sections s and t of A (ie, s, t ∈ Γ(U, A) ≡ A(U ), with U ? X open). It can be shown that the usual di?erential operator ? in (1) above is the prototype of a ?at A-connection [67, 68]. The aforementioned generalization of the usual di?erential operator ? to an (abstract) Aconnection D involves two steps emulating the de?nition of ? above. First, one identi?es D with a suitable (C-linear) sheaf morphism as in (1), and second, one secures that the Leibniz condition is satis?ed by D , as in (2) above. So, given a di?erential triad T = (A, ?, ?), let E be an A-module sheaf on X . Then, the ?rst step corresponds to de?ning D as a map D : E ?→ E ?A ? ? = ? ?A E ≡ ?(E ) (3)

for α ∈ A(U ), s ∈ E (U ) ≡ Γ(U, E ), and U open in X . The connection D as de?ned above may be coined a ‘Koszul linear connection’ and its existence on the vector sheaf E is crucially dependent on both the base space X and the structure sheaf A. For X a paracompact and Hausdor? topological space, and for AX a ?ne sheaf on it, the existence of D is well secured, as for instance in the case of C ∞ -smooth manifolds [67, 68]. 2.1.1 The local form of D

Given a local gauge eU ≡ {U ; (ei )0≤i≤n?1 } of the vector sheaf E of rank n,11 every continuous local section s ∈ E (U ) (U ∈ U ) can be expressed as a unique superposition n i=1 si ei with coe?cientsThe pair (X, AX ) is called a C-algebraized space, where C corresponds to the constant sheaf of the complex ? numbers C over X , which is naturally injected into AX (ie, C → AX and, plainly, C = Γ(X, C) ≡ C(X )). It is tacitly assumed that for every open set U in X , the algebra A(U ) of continuous local sections of AX is a unital, commutative and associative algebra over C. It must be noted here however that one could start with a K-algebraized space (K = R, C) in which the structure sheaf AX would consist of unital, abelian and associative algebras over the ?elds K = R, C respectively. Here we have just ?xed K to the complete ?eld of complex numbers, but in the future we are going to discuss also the real case. Also, in either case AX is assumed to be ?ne. In the sequel, when it is rather clear what the base topological space X is, we will omit it from AX and simply write A. 11 We recall from [67, 68, 75] that in ADG, U = {Uα }α∈I is called a local frame or a coordinatizing open cover of, or even a local choice of basis (or gauge!) for E . The ei s in eU are local sections of E (ie, elements of Γ(U, E )) constituting a basis of E (U ). We also mention that for the A-module sheaf E , regarded as a vector sheaf of rank n, one has by de?nition the following A|U -isomorphisms: E|U = An |U = (A|U )n and, concomitantly, the following10

So that, in toto, every connection D can be written locally as D =?+ω (8)

with (8) e?ectively expressing the procedure commonly known in physics as localizing or gauging the usual (?at) di?erential ? to the (curved) covariant derivative D . Thus, the (non-?at) ω part of D , called the gauge potential in physics, measures the deviation from di?erentiating ?atly (ie, by ? ), when one di?erentiates ‘covariantly’ by D .12 2.1.2 Local gauge transformations of D

We investigate here, in the context of ADG, the behavior of the gauge potential part A of D under local gauge transformations—the so-called ‘transformation law of potentials’ in [67, 68]. Thus, let E be an A-module or a vector sheaf of rank n. Let eU ≡ {U ; ei=1···n } and f V ≡ {V ; fi=1···n } be local gauges of E over the open sets U and V of X which, in turn, we assume have non-empty intersection U ∩ V . Let us denote by g ≡ (gij ) the following change of local gauge matrixn

fj =i=1n n

gij ei

(9)

equalities section-wise: E (U ) = A (U ) = A(U ) (with An the n-fold Whitney sum of A with itself). Thus, E is a locally free A-module of ?nite rank n—an appellation synonymous to vector sheaf in ADG [67, 68]. For n = 1, the vector sheaf E is called a line sheaf and it is symbolized by L. 12 In the sequel we will symbolize the gauge potential part of D in (8) by A instead of ω in order to be in agreement with our notation in the previous papers [74, 75], as well as with the standard notation for the spinLorentzian connection in current Lorentzian quantum gravity research [6, 7, 8, 14].

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which, plainly, is a local (ie, relative to U ∩ V ) section of the ‘natural’ structure group sheaf GL(n, A) of E 13 —that is, gij ∈ GL(n, A(U ∩ V )) = GL(n, A)(U ∩ V ). Without going into the details of the derivation, which can be found in [67, 68], we note that under such a local gauge transformation g , the gauge potential part ω ≡ A of D in (8) transforms as follows A = g ?1 Ag + g ?1?g′

(10)

a way we are familiar with from the usual di?erential geometry of the smooth ?ber bundles of gauge theories. For completeness, it must be noted here that, in (10), A ≡ (Aij ) ∈ Mn (?1 (U )) = ′ ′ Mn (?1 )(U ) and A ≡ (Aij ) ∈ Mn (?1 (V )) = Mn (?1 )(V ). The transformation of A under local gauge changes is called a?ne or inhomogeneous in the usual gauge-theoretic parlance precisely because of the term g ?1?g . We will return to this a?ne term in subsection 2.3 and subsequently in section 5 where we will comment on the essentially non-geometrical (ie, non-tensorial) character of connection. Also, anticipating our discussion of moduli spaces of gauge-equivalent connections g in the next section, we note that (10) expresses an equivalence relation ‘?’ between the gauge ′ potentials A and A .

2.2

Pseudo-Riemannian (Lorentzian) Metric Connections

In this subsection we are interested in endowing a vector sheaf E of ?nite rank n ∈ N with an inde?nite A-valued symmetric inner product ρ, and, concomitantly, study A-connections D that are compatible with the (inde?nite) metric g associated with ρ—the so-called metric connections. With an eye towards the applications to Lorentzian (quantum) gravity in the sequel, we are particularly interested in metric D s relative to Lorentzian metrics of signature diag(g ) = (?, +, +, · · · ). Also, continuing our work [74] which dealt with principal Lorentzian ?nsheaves of qausets, we are interested in the group sheaves AutA(E ) of A-automorphisms of E —the principal sheaves of structure symmetries of E .14 In the case of a real (ie, K = R and R-algebraized space) Lorentzian vector sheaf (E , ρ) of rank 4,15 the stalks of the corresponding G -sheaves will ‘naturally’We will present some rudiments of structure group (or principal or G -) sheaves of associated vector sheaves E in the next subsection. One may recognize GL(n, A) above as the local version of the automorphism group sheaf AutE of E . The adjective ‘local’ here pertains to the fact mentioned earlier that ADG assumes that E is locally isomorphic to An . 14 Commonly known as G -sheaves in the mathematical literature [67]. 15 We would like to declare up-front that in this paper we provide no argument whatsoever for assuming that the dimensionality (rank) n of our vector sheaves is the ‘empirical’ (or better, ‘conventional’) 4 of the spacetime manifold of ‘macroscopic experience’ (or better, of the classical theory). In the course of this work the reader will realize that all our constructions are manifestly independent of the classical 4-dimensional, locally Euclidean, C ∞ -smooth, Lorentzian spacetime manifold of general relativity so that we will time and again doubt whether the latter, and the host of (mathematical) structures that classically it is thought of as carrying (eg, its uncountably in?nite cardinality of events, its dimensionality, its topological, di?erential and metric structures), is a physically meaningful concept. For example, we will maintain that dimensionality and the metric are free mathematical choices of (ie, ?xed by) the theorist and not Nature’s own, while that the topology and di?erential structure are inherent in the dynamical objects (?elds) that may be thought of as living and propagating on ‘spacetime’, not by that inert background ‘spacetime’ itself, which is devoid of any physical meaning. Moreover, all this will be13

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11

be assumed to host the group SO (1, 3)↑—the orthochronous Lorentz group of (local) isometries of (E , ρ) which, in turn, is locally isomorphic to the spin-group SL(2, C).16 We thus catch a ?rst glimpse of the spin-Lorentzian connections considered in the context of curved ?nsheaves of qausets in [74], which will be dealt with in more detail in section 4. Thus, let E be a vector sheaf. By an A-valued pseudo-Riemannian inner product ρ on E (over X ) we mean a sheaf morphism ρ : E ⊕ E ?→ A (11)

which is i) A-bilinear between the A-modules concerned, ii) symmetric (ie, ρ(s, t) = ρ(t, s), s, t ∈ E (U )) and of inde?nite signature, as well as iii) strongly non-degenerate. That is, we assume that ρ(s, t), for any two local sections s and t in E (U ),17 is given via the canonical isomorphism E? = E? between E and its dual E ? , as ρ ?(s)(t) := ρ(s, t) (13)ρ ?

(12)

with (12) being true up to an A-isomorphism18 . We further assume that for the vector sheaf E (of ?nite rank n ∈ N) endowed with the Aconnection D , the vector sheaf ? in the given di?erential triad T = (A, ?, ?) is the dual of E appearing in (12) (ie, ? = E ? ≡ HomA (E , A)). Thus, in line with the usual Christo?el theory [67, 68], we can de?ne a linear connection ?, as follows ? : E × E ?→ E (14)

expressed in an algebraic, locally ?nite setting quite remote from the uncountable continuous in?nity of events of the manifold. 16 In the sense that their corresponding Lie algebras are isomorphic: so(1, 3)↑ ? sl(2, C) [74]. 17 It is important to notice here that the A-metric ρ is not a (bilinear) map assigned to the points of the base space X per se (which is only assumed to be a topological, not a di?erential, let alone a metric, space), but to the ?bers (stalks) of the relevant module or vector sheaves which are inhabited by the geometrical objects that live on X . As noted in a previous footnote, in our scheme, metric and, as we shall see later, topological and di?erential properties concern the objects that live on ‘space(time)’, not the supporting space(time) itself. This recalls Gauss’ and Riemann’s original labors with endowing the linear ?ber spaces tangent to a sphere with a bilinear quadratic form—a metric. They ascribed a metric to the linear ?bers, not to the supporting sphere itself which, anyway, is manifestly ‘non-linear’ [73]. What we wish to highlight by these remarks is that space(time) carries no metric. Equally important is to note that the A-valued metric ρ is imposed on these objects by us and it is intimately tied to (ie, takes values in) our own measurements (arithmetics) in A (see comparison between the notions of connection and curvature in 2.3.5). ρ is not a property of space(time), which does not exist (in a physical sense) anyway; rather, it is an attribute related to our own measurements of ‘it all’. These remarks are important for our subsequent physical interpretation of ADG in its application to ?nitary Lorentzian quantum gravity in the next four sections. It is a preliminary indication that in our theory the base space(time) is an ether-like ‘substance’ without any physical signi?cance. See remarks about ‘gravity as a gauge theory’ in the next section, about the ‘physical insigni?cance’ or ‘non-physicality’ of spacetime in 5.1.1 and about ‘the relativity of di?erentiability’ in 6.2, as well as some similar anticipations in [74, 75]. 18 The epithet ‘strongly’ to ‘non-degenerate’ above indicates that ρ ? in (12) is also onto.

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acting section-wise on E (U ) as ?(s, t) ≡ ?s (t) := D (t)(s) (15)

Now, one says that D is a pseudo-Riemannian A-connection or that it is compatible with the inde?nite metric g of the inner product ρ in (11), whenever it ful?lls the following two conditions: ? Riemannian symmetry: ?(s, t) ? ?(t, s) = [s, t]; for s, t ∈ E (U ) and [ . , . ] the usual Lie bracket (product). ? Ricci identity: ? (ρ(s, t))(u) = ρ(?(u, s), t) + ρ(s, ?(u, t)); for s, t, u ∈ E (U ), as usual. In particular, for a Lorentzian ρ and its associated g ,19 an A-connection D is said to be compatible with the Lorentz A-inner product ρ on E 20 when its associated Christo?el ? in (14) satis?es ?ρ = 0 (16)

It is worth reminding the reader who is familiar with the usual theory that (17) above implies that the Levi-Civita A-connection D induced by the Lorentz A-metric ρ is torsion-free [70]. 2.2.1 Connections on (Lorentzian) principal sheaves

As mentioned in the beginning of this subsection, of special interest in our study is the case of a (real) Lorentzian vector sheaf (E , ρ) of rank 4 whose A-automorphism sheaf AutA E ↑ bears G = L↑ := SO (1, 3)↑—the orthochronous ρ-preserving A-automorphisms of E in its stalks.21 L+ is the principal sheaf of structure symmetries of E ↑ . In turn, E ↑ is called the L+ -associated vector sheaf.22 But let us ?rst give a brief discussion of connections on principal sheaves ` a la ADG and then focus on spin-Lorentzian (metric) connections. The reader will have to wait until section 4 whereWith respect to a local (coordinate) gauge eU ≡ {U ; (ei )0≤i≤n?1 } of the vector sheaf E of rank n, ρ(ei , ej ) = gij = diag(?1, +1, · · · ) [67, 68]. 20 Such a metric connection is commonly known as Levi-Civita connection. 21 One may wish to symbolize the pair (E , ρ) by E ↑ , thus AutA E ↑ by L+ . In the sequel, when it is clear from the context that we are talking about a Lorentzian vector sheaf E ↑ = (E , ρ), we may use the symbols E and E ↑ for it interchangeably hopefully without confusion. For a general vector sheaf E , AutA E is a subsheaf of E ndE , ? in fact, for a given open U ? X , AutA (E )(U ) ? EndA (E|U ) —the upper dot denoting invertible endomorphisms. ? We thus write in general: AutA (E ) ≡ AutE := (E ndE ) . 22 Henceforth we will assume that every principal sheaf acts on the typical stalk of its associated sheaf on the left (see below).19

It must be noted here that in the same way that ADG—the di?erential geometry of vector sheaves—represents an abstraction and generalization of the usual calculus on vector bundles over C ∞ -smooth manifolds to the e?ect that no calculus, in the usual sense, is employed at all [67, 68], Lie sheaves of groups are the abstract analogues of the usual Lie groups that play a central role in the classical di?erential geometry of principal ?ber bundles over di?erential manifolds [124, 125, 126]. Thus, let G be a Lie sheaf of groups as above. Formally speaking, by a principal sheaf P with ˙ )27 we mean a quadruple (P , L, X, π ) consisting of a structure group G relative to G = (L, E , σ, ? 28 sheaf of sets P such that:By abuse of notation, and hopefully without confusing the reader, in the sequel we will also symbolize the groups that dwell in the stalks of G by ‘G ’. 24 The reader should note that in the present paper we symbolize the gauge (structure) group of both Y-M theory and gravity also by G , hopefully without causing any confusion between it and the abstract Lie sheaf of groups above. 25 By assuming that the group sheaf G in (18) is a sheaf of Lie groups, we may take L to be the corresponding sheaf of Lie algebras. 26 ˙ ? is also known as the logarithmic di?erential of G . 27 Where L is the sheaf of Lie algebras of the Lie group sheaf G . L is supposed to represent the local structural type of P [125]. 28 P may be thought of as ‘coordinatizing’ the principal sheaf, thus we use the same symbol ‘P ’ for the principal sheaf and its coordinatizing sheaf of sets. π is the usual projection map from P to the base space X . For more details, refer to [124, 125, 126].23

in complete analogy to the local gauge potential 1-forms A of connections D on vector sheaves presented in (5)–(8).31Otherwise called the P -, or even, the L-associated vector sheaf. ˙ : P ?→ ?1 ?A L(≡ ?1 (L)), to manifest the usual statement This morphism can be equivalently written as D that a connection on a principal sheaf is a Lie algebra-valued 1-form. Time and again we will encounter this de?nition below. 31 ˙ s obey a transformation law of Furthermore, one can show that for a local change of gauge g as in (9), the A potentials completely analogous to the one obeyed by the As in (10). Without going into any details, it reads: ˙ (σ (g ?1 ) ≡ σ (g )?1 ) [124, 125, 126]. ˙ ′ = σ (g )?1 A ˙ σ (g ) + σ (g )?1 ?g, A30 29

Finitary, Causal and Quantal Vacuum Einstein Gravity

15

˙ on principal sheaves P in Now, the essential point in this presentation of connections D relation to our presentation of A-connections D on vector sheaves E earlier, is that when the latter are the P -associated sheaves relative to corresponding representations σ : L ?→ AutE , the following ‘commutative diagram’ may be used to picture formally the ‘σ -induced projection ˙ on P to D on E σ ? ’ of D P ˙ D ?1 (L)c E

σ ? E A D id ?1 ( E )c

(29)

where σ ? may be regarded a morphism between P and A regarded simply as sheaves of structureless sets.32 To make an initial contact with [74], we can now particularize the general ADG-based presentation of principal sheaves P above to (real) Lorentzian G -sheaves. As brie?y noted earlier, the structure group G dwelling in the stalks of the latter is taken to be L↑ := SO (1, 3)↑—the Lie group of orthochronous Lorentz A-isometries, so that P in this case is denoted by L+ . The L+ -associated sheaf E ↑ = (E , ρ) is a (real) vector sheaf of rank 4, equipped with an A-metric ρ of absolute trace equal to 2. Thus, there is a local homomorphism (representation) σ of the Lie algebra so(1, 3)↑ ? sl(2, C) of the structure group L↑ in L+ into the ‘Lie algebra’ sheaf autA (E ↑) of the group sheaf AutA (E ↑ ) of invertible A-endomorphisms of E preserving the Lorentzian A-metric ρ—that is, the A-metric ρ symmetries (isometries) of E ↑ . Collecting information from our presentation of connections on G -sheaves and their associated vector sheaves, we are in a position now to recall from [74] that, in the particular case of the L+ -associated vector sheaf E ↑ , the gauge potential part A of an A-connection D on E ↑ is an so(1, 3)↑ ? sl(2, C)valued 1-form on L+ . the so-called spin-Lorentzian connection 1-form. After we discuss the a?ne space A of Y-M and Lorentzian gravitational G -connections from an ADG-theoretic perspective in 2.4, as well as present the connection-based vacuum Einstein equations ADG-theoretically in the next section, we are going to return to the kinematical spinLorentzian connections on principal ?nsheaves of qausets and their associated vector sheavesThat is to say, by forgetting both the group structure of the G -sheaf P and the algebra structure of the ˙ on it from its structure sheaf A. The inverse procedure of building the principal sheaf P and the connection D associated vector sheaf E and the connection D on it, may be loosely called ‘σ -induced lifting σ ? ?1 ’ of (E , D) to ˙ ). The σ ?1 -lifting is a forgetful correspondence since, in going from a vector sheaf to its structure group (P , D sheaf, the linear structure of the former is lost—something which is in fact re?ected on that, while D is C-linear, ˙ is not. However, for more details about commutative diagrams like (29) between principal sheaves (P1 , D ˙ 1 ) and D ˙ 2 ), their corresponding associated sheaves (E1 , D1 ) and (E1 , D1 ), as well as the respective projections σ (P2 , D ? of the former to the latter, the reader is referred to [126].32

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studied in [74] in section 4, then we will formulate their dynamical vacuum Einstein equations in 5, and ?nally, in the same section, we will discuss a possible covariant (ie, action-based, path integral-type of) quantum dynamics for them.

Therefore, any time we have the C-linear morphism D and its prolongation D 1 at our disposal, we can de?ne the curvature R(D ) of the connection D .34 By de?ning a curvature space as the ?nite sequence (A, ?, ?1 , d, ?2 ) of A-modules and C-linear morphisms between them, we can distill the last statement to the following: we can always de?ne the curvature R of a given A-connection D , provided we have a curvature space. As a matter of fact, it is rather straightforward to see that, for E a vector sheaf, R(D ) is an A-morphism of A-modules, in the following sense R ∈ HomA (E , ?2 (E )) = HomA (E , ?2(E ))(X ) ?2 (E ndE )(X ) = Z 0 (U , ?2 (E ndE )) (37)

where U = {Uα }α∈I is an open cover of X and Z 0 (U , ?2 (E ndE )) the A(U )-module of 0-cocycles of ?2 (E ndE ) relative to the U -coordinatization of X .35 2.3.1 The local form of R

Motivated by (37) and the last remarks, we are in a position to give the local form for the curvature R of a given A-connection D . Thus, let E be a vector sheaf of rank n, D an Aconnection on it and U = {Uα }α∈I a local coordinatization frame of it. By virtue of (37) we haveα R(D ) = R = (Rij ) ≡ ((Rij )) ∈ Z 0 (U , ?2 (E ndE )) ? α ?2 (E ndE )(Uα ) = α Mn (?2 (Uα )) (α)

(38)

so that we are led to remark that: the curvature R of an A-connection D on a vector sheaf E of rank n is a 0-cocycle of local n × n matrices having for entries local sections of ?2 —ie, local 2-forms on X .In connection with (36), one can justify our earlier remark that the standard di?erential operator ? , regarded as an A-connection as in (1) (ie, as the sheaf morphism ? : A ?→ ?1 = A ?A ?1 ≡ ?1 (A)), is ?at, since: R(? ) = d?? = d1 ?d0 ≡ d2 = 0 (which is secured by the nilpotency of the usual Cartan-K¨ ahler (exterior) di?erential operator d [75]). In the latter paper, and in a sheaf-cohomological fashion, it was shown that it is exactly D’s deviation from nilpotency (ie, from ?atness), which in turn de?nes a non-vanishing curvature R(D) = D2 = 0, that prevents a sequence · · · ?→ ?i ?→ ?i+1 ?→ · · · of di?erential A-module sheaves ?i and C-linear sheaf morphisms Di between them from being a complex. (Di , i ≥ 2, stand for high-order prolongations of the D0 ≡ D and D1 connections above [67, 68].) 35 One may wish to recall that, for a vector sheaf E like the one involved in (37), E ndE ≡ HomA (E , E ) ? = E ?A E ? = E ? ?A E .D i?1 Di D i+1 34

18 2.3.2

Anastasios Mallios and Ioannis Raptis

Local gauge transformations of R

We wish to investigate here the behavior of the curvature R(D ) of an A-connection D under local gauge transformations—the so-called ‘transformation law of ?eld strengths’ in the usual gauge-theoretic parlance and in ADG [67, 68]. Thus, let g ≡ gij ∈ GL(n, A)(U ∩ V ) be the change-of-gauge matrix we considered in (9) in connection with the transformation law of gauge potentials. Again, without going into the details of the derivation, we bring forth from [67, 68] the following local transformation law of gauge ?eld strengths for a local frame change : eU ?→ eV (U, V open gauges in X ), g ′ the curvature transforms as : R ?→ R = g ?1Rgg

(39)

which we are familiar with from the usual di?erential geometric (ie, smooth ?ber bundle-theoretic) treatment of gauge theories. For completeness, we remind ourselves here that, in (39), RU ∩V ≡ U ∩V (Rij ) ∈ Mn (?2 (U ∩ V ))—an n × n matrix of sections of local 2-forms. The transformation of R under local gauge changes is called homogeneous or covariant in the usual gauge-theoretic parlance. We will return to this term in 2.3.5 and subsequently in section 5 where we will comment on the geometrical (ie, tensorial) character of curvature. 2.3.3 Cartan’s structural equation—Bianchi identities

As a result of the extension of d to d, the aforementioned curvature space (A, ?, ?1 , d, ?2 ), when enriched with the A-module sheaf ?3 as well as with the nilpotent C-linear morphism d in (44), becomes a so-called Bianchi space. In a Bianchi space, the usual second Bianchi identity holds dR ≡ dR = [R, A] ≡ R ∧ A ? A ∧ R (47)

where d is understood to e?ect coordinate-wise: d : Mn (?2 ) ?→ Mn (?3 ). In the case of a line sheaf L, one can easily show by using (30) and the nilpotency of d that dR = 0 (48)

which is usually referred to as the homogeneous ?eld equation. The latter, in turn, translates to the following cohomological statement: the curvature R of an A-connection D on a line sheaf L over X provides a closed 2-form on X . which came very handy in the sheaf-cohomological classi?cation of the curved associated line sheaves of qausets and their quanta—the so-called ‘causons’—performed in [75]. 2 Finally, one can also show that the second prolongation DE ndE of the induced A-connection ? ? DE ndE on E ndE = E ?A E satis?es the following ‘covariant version’ of the second Bianchi identity (47) above2 DE ndE (R) = 0 2 2 3 where DE ndE : ? (E ndE ) ?→ ? (E ndE ). Thus, similarly to (47), one also shows that

(49)

DE ndE R = dR + [A, R]36

(50)

In the sequel, following the cohomological custom in [75], we identify ? , d and d (and all higher order exterior derivations) with the generic Cartan di?erential d, specifying its order only when necessary and by writing generically di (i ≥ 0).

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which proves the equivalence of the second (exterior di?erential) Bianchi identity on E and its induced (covariant di?erential) version on E ndE . 2.3.4 The Ricci tensor, scalar and the Einstein-Lorentz (curvature) space

which, plainly, is A(U )-valued. We have built a suitable conceptual background to arrive now at a central notion in this paper. A (real) Lorentzian vector sheaf E ↑ = (E , ρ) over a R-algebraized space (X, A) such that: 1. it is supported by a di?erential triad T = (A, ?, ?1 ) relative to which (12) holds, that is, E ? ≡ ?1 , 2. there is a R-linear Lorentzian connection D on it satisfying (17) (ie, a metric connection) and, furthermore, 3. it is a curvature space (A, ?, ?1 , d, ?2 ) supporting a null R, that is to say, a Ricci scalar operator satisfying the vacuum Einstein equations R(E ) = 0 (53)

is called an Einstein-Lorentz (E-L) space, while the corresponding base space X , an Einstein space [70].39 Of course, it has been implicitly assumed that, for an appropriate choice of structure sheaf A, equation (53) can be actually derived from the variation of the corresponding Lagrangian density (alias, Einstein-Hilbert action functional EH). We will return to this assumption in the next section. In connection with the de?nition of an Einstein space X , it is worth noting thatThe reader should note that below, and only in the vacuum Einstein case, we will symbolize the connections involved by D instead of the calligraphic D we have used so far to denote the general A-connections in ADG. 38 Due to this, R has been called a curvature endomorphism in [70]. 39 In the next section, where we will cast Lorentzian gravity as a Y-M-type of gauge theory a ` la ADG, we will also de?ne a Yang-Mills space analogous to the Einstein space above.37

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21

the only structural requirement that ADG places on the Einstein base space X is that it is, merely, a topological space—in fact, an arbitrary topological space, without any assumptions whatsoever about its di?erential, let alone its metric, structure. This prompts us to emphasize, once again [67, 68, 70, 77, 78, 74, 75, 73], the essential ‘working philosophy’ of ADG: to actually do di?erential geometry one need not assume any ‘background di?erentiable space’ X , for di?erentiability derives from the algebraic structure of the objects (structure algebras) that live on that ‘space’. The only role of the latter is a secondary, auxiliary and, arguably, a ‘physically atrophic’ one in comparison to the active role played by those objects (in particular, the algebra A(U ) of local sections of A) themselves: X merely provides an inert, ether-like sca?olding for the localization and the dynamical interactions (‘algebraically and sheaf-theoretically modelled interrelations’) of those physically signi?cant objects—a passive substrate of no physical signi?cance whatsoever, since it does not actively participate into the algebraico-dynamical relations between the objects themselves.40 All in all, the basic objects that ADG works with is the sections of the sheaves in focus—that is, the entities that live in the stalks of the relevant sheaves, and not with the underlying base space X , so that any notion of ‘di?erentiability’ according to ADG derives its sense from the algebraic relations between (ie, the algebraic structure of ) those (local) sections, with the apparently ‘intervening between’ or ‘permeating through these objects’ background space X playing absolutely no role in it. 2.3.5 A fundamental di?erence between D and R(D ) and its physical interpretation

At this point it is worth stressing a characteristic di?erence between an A-connection D and its curvature R(D )—a di?erence that is emphasized by ADG, it has a signi?cant bearing on the physical interpretation of our theory, and it has been already highlighted in both [74] and [75]; namely that, while R is an A-morphism, D is only a K-morphism (K = R, C). This means that, since the structure sheaf A corresponds to ‘geometry’ in our algebraic scheme, in the sense that A(U )—the algebra of local sections of A—represents the algebra of local operationsIts arbitrary character—again, X is assumed to be simply an arbitrary topological space—re?ects precisely its physical insigni?cance. This non-physicality, the ‘algebraic inactivity’ and ‘dynamically non-participatory character’ so to speak, of the background space will become transparent subsequently when we formulate the dynamical equations for vacuum gravity entirely in terms of sheaf morphisms between the objects—ie, virtually the sections—that live on X (the main sheaf morphism being the connection D—arguably the central operator with which one actually does di?erential geometry!). At this point we would like to further note, according to [67], that a sheaf morphism is actually reduced to a family of (local) morphisms between (the complete presheaves of) local sections M or(E , F ) ? φ ←→ (φU ) ∈ M or(Γ(E ), Γ(F )—a category equivalence through (the section functor) Γ. In the last section we will return to the inert, passive, ether-like character of the base space in the particular case that X is (a region of) a C ∞ -smooth spacetime manifold. There we will argue how ADG ‘relativizes’ the ‘di?erential properties’ of space(time).40

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of measurement (of the quantum system ‘space-time’) relative to the local laboratory (frame, or gauge, or even ‘observation device’) U [87, 74, 75], it e?ectively encodes our geometrical information about the physical system in focus.41 Consequently, R, which, being an A-morphism, respects our local measurements—the ‘geometryencoding (measuring) apparatus’ A of ADG so to speak—is a geometrical object (ie, a tensor) in our theory and lies on the classical side of the quantum divide. On the other hand, D , which respects only the constant sheaf K(= R, C) but not our (local) measurements in A, is not a geometrical object42 and it lies on the quantum (ie, the purely algebraic, ` a la Leibniz [73]), side of Heisenberg’s cut.43

2.4

The A?ne Space A of A-Connections

We ?x the K-algebraized space (X, A) and the di?erential triad T = (A, ?, ?) on it with which we are working, and we let E be an A-module on X . We denote by AA (E ) (54)

the set of A-connections on E . By de?nition (3), AA (E ) is a subset of HomK (E , ?(E )) (? ≡ ?1 ) whose zero element may be regarded as the zero A-connection in AA (E ). However, by (4), one infers that ? is also zero in this case, thus we will exclude altogether the zero A-connection fromAs mentioned before, AX is the abelian algebra sheaf of ‘generalized arithmetics’ in ADG generalizing the ∞ usual commutative coordinate sheaf R CM of the smooth manifold—the sheaf of abelian rings R C ∞ (M ) of in?nitely di?erentiable, real-valued functions on the di?erential manifold M . We tacitly assume in our theory that ‘geometry’ is synonymous to ‘measurement’; hence, in the quantum context, it is intimately related to ‘observation’ (being, in fact, the result of it). Furthermore, since the results of observation arguably lie on the classical side of the quantum divide (the so-called Heisenberg Schnitt), A must be a sheaf of abelian algebras. This is supposed to be a concise ADG-theoretic encodement of Bohr’s correspondence principle, namely, that the numbers that we obtain upon measuring the properties of a quantum mechanical system (the so-called q -numbers) must be commutative (the so-called c-numbers). In other words, the acts of measurement yield c-numbers from q -numbers, so that ‘geometry’—the structural analysis of (the algebras of our local measurements of) ‘space’—deals, by de?nition, with commutative numbers and the (sheaves of) abelian algebras into which the latter are e?ectively encoded. See also closing remarks in [68] for a similar discussion of ‘geometry a ` la ADG’ in the sense above, as well as our remarks about Gel’fand duality in 5.5.1 42 Another way to say this is that the notion of connection is algebraic (ie, analytic), not geometrical. In short, D is not a tensor. That R is a tensor while D is not is re?ected in their (local) gauge transformation laws that we saw earlier: A transforms a?nely or inhomogeneously (non-tensorially), while R covariantly or homogeneously (tensorially) under a (local) change of gauges. 43 Although it must be also stressed that D, like the usual notion of derivative ? that it generalizes, has a geometrical interpretation. As the derivative of a function (of a single variable) is usually interpreted in a Newtonian fashion as the slope (gradient) of the tangent to the curve (graph) of the function, so D can be interpreted geometrically as a parallel transporter of objects (here, A-tensors) along geometrical curves (paths) in space(time). However, it is rather inappropriate to think of D as a geometrical object proper and at the same maintain a geometrical interpretation for it, for does it not sound redundant to ask for the geometrical interpretation of an ‘inherently geometrical’ object, like the triangle or the circle, for instance? In other words, if the notion of connection was ‘inherently geometrical’, it would certainly be super?uous to also have a geometrical interpretation for it.41

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AA (E ). Since any connection may be taken to serve as an ‘origin’ for the space of A-connections, we conclude that AA (E ) is an a?ne space modelled after the A(X )-module HomK (E , ?(E )). For a vector sheaf E , HomK (E , ?(E )) becomes ?(E ndE )(X ). Now, in connection with the statement above, let D be an A-connection in AA (E ) ≡ HomK (E , ?(E )). ′ Then, it can be shown [67, 68] that any other connection D in AA (E ) is of the form D =D+u′

(55)

for a uniquely de?ned u ∈ HomA (E , ?1 (E )). For E a vector sheaf, u belongs to ?1 (E ndE )(X ). Thus, for a given D ∈ AA (E ) we can formally write (55) as: AA (E ) = D + HomA (E , ?1 (E )), within a bijection. Interestingly enough, (55) tells us that the di?erence of two connections, which are K-linear sheaf morphisms, is an A-morphism like the curvature; hence, in view of the ′ comparison between D and R(D ) above, we can say that D ? D is a geometrical object since it respects our measurements in A by transforming homogeneously (tensorially) under (local) gauge transformations.44 In the particular case of a line sheaf L, AA (L) can be identi?ed with ?1 (X )—the A(X )-module of ‘1-forms’ on X . Thus, given any connection D in AA (L), any other connection D on L can be written as D = D + ω for some unique ω in ?1 (X ). This result was used in [75] for the sheaf-cohomological classi?cation of the line sheaves associated with the curved principal ?nsheaves of qausets and the non-trivial connections on them in [74]. We will return to AA (E ) in the next section where we will factor it by the structure (gauge) group G = Aut(E ) of E to obtain the orbifold or moduli space AA (E )/G of gauge-equivalent connections on E of a Y-M or gravitational type depending on G .′ ′

3

Vacuum Einstein Gravity as a Y-M-type of Gauge Theory ` a la ADG

In this section we present the usual vacuum Einstein gravity in the language of ADG, that is, as a Y-M-type of gauge theory describing the dynamics of a Lorentzian connection on a suitable principal Lorentzian sheaf and its associated vector sheaf, in short, on an E-L space as de?ned above. We present only the material that we feel is relevant to our subsequent presentation of ?nitary vacuum Lorentzian gravity encouraging the reader to refer to [67, 68, 70, 71] for more analytical treatment of Y-M theories and gravity ` a la ADG. But let us ?rst motivate in a rather general way this conception of gravity as a gauge theory.44

The reader could verify that u transforms covariantly under (local) changes of gauge.

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3.1

Physical Motivation

It is well known that the original formulation of general relativity was in terms of a pseudoRiemannian metric g?ν on a C ∞ -smooth spacetime manifold M . For Einstein, the ten components of the metric represented the gravitational potentials—the pure gravitational dynamical degrees of freedom so to speak. However, very early on it was realized that there was an equivalent formulation of general relativity involving the dynamics of the so-called spin-connection ω . This approach came to be known as Einstein-Cartan theory [46] and arguably it was the ?rst indication, long before the advent of the Y-M gauge theories of matter, that gravity concealed some sort of gauge invariance which was simply masked by the metric formulation.45 In fact, Feynman, in an attempt to view gravity purely ?eld-theoretically and, in extenso, quantum gravity as a quantum ?eld theory (ie, in an attempt to quantize gravity using a language and techniques more familiar to a particle physicist than a general relativist,46 ) he essentially ‘downplayed’, or at least undermined, the di?erential geometric picture of general relativity and instead he concentrated on its gauge-theoretic attributes. Brian Hat?eld nicely reconstructed Feynman’s attitude towards (quantum) gravity in [40],47 as follows“...Thus it is no surprise that Feynman would recreate general relativity from a nongeometrical viewpoint. The practical side of this approach is that one does not have to learn some ‘fancy-schmanzy’ (as he liked to call it) di?erential geometry in order to study gravitational physics. (Instead, one would just have to learn some quantum ?eld theory.) However, when the ultimate goal is to quantize gravity, Feynman felt that the geometrical interpretation just stood in the way. From the ?eld theoretic viewpoint, one could avoid actually de?ning—up front—the physical meaning of quantum geometry, ?uctuating topology, space-time foam, etc., and instead look for the geometrical meaning after quantization...Feynman certainly felt that the geometrical interpretation is marvellous, but ‘the fact that a massless spin-2 ?eld can be interpreted as a metric was simply a coincidence that might be understood as representing some kind of gauge invariance’.”48

Feynman’s ‘negative’ attitude towards the standard di?erential geometry and the smooth spacetime continuum that supports it,49 especially if we consider the unrenormalizable in?nities that plague quantum gravity when treated as another quantum ?eld theory, is quite understandable if we recall from the beginning of the present paper his earlier position—repeated once again,Recently, after reading [60], the present authors have become aware of a very early attempt by Eddington at formulating general relativity (also entertaining the possibility of unifying gravity with electromagnetism) based solely on the a?ne connection and not on the metric, which is treated as a secondary structure, ‘derivative’ in some sense from the connection. Indicatively, Kostro writes: “...[Eddington’s] approach relied on a?ne geometry. In this geometry, connection, and not metric, is considered to be the basic mathematical entity. The metric g?ν (x) needed for the description of gravitational interactions, appears here as something secondary, which is derived from connection...” (bottom of page 99 and references therein). 46 Such an approach was championed a decade later by Weinberg in a celebrated book [127]. 47 See Hat?eld’s preamble titled ‘Quantum Gravity’. 48 Our emphasis of Feynman’s words as quoted by Hat?eld. 49 The reader must have realized by now that by the epithets ‘standard’, or ‘usual’, or more importantly, ‘classical’, to ‘di?erential geometry’ we mean the di?erential geometry of C ∞ -smooth manifolds—the so-called ‘calculus on di?erential manifolds’.45

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25

that “the theory that space is continuous is wrong, because we get...in?nities...the simple ideas of geometry, extended down to in?nitely small, are wrong!”.50 However, it must be noted that Feynman’s ‘unconventional’ attempt in the early 60s to tackle the problem of quantum gravity gauge quantum ?eld-theoretically was preceded by Bergmann’s ingenious recasting of the Einstein-Cartan theory in terms of 2-component spinors, thus e?ectively showing that the main dynamical ?eld involved in that theory—the spin connection ω —is an sl(2, C)-valued 1-form [16].51 All in all, it is remarkable indeed that such a connectionbased approach to general relativity, classical or quantum, has been revived in the last ?fteen years or so in the context of non-perturbative canonical quantum gravity. We refer of course to Ashtekar’s modi?cation of the Palatini vierbein or comoving 4-frame-based formalism by using new canonical variables to describe the phase space of general relativity and in which variables the gravitational constraints are signi?cantly simpli?ed [3]. Interestingly enough, and in relation to Bergmann’s work mentioned brie?y above, in Ashtekar’s scheme the principal dynamical variable is an sl(2, C)-valued self-dual spin-Lorentzian connection 1-form A+52 [3]. But after this lengthy preamble let us get on with our main aim in this section to present the classical vacuum Lorentzian gravity as a Y-M-type of gauge theory in the manner of ADG.

3.2

Y-M Theory ` a la ADG—Y-M Curvature Space

Let (E , ρ) be a (real) Lorentzian vector sheaf of ?nite rank n associated with a di?erential triad T = (A, ?, ?1 ), which in turn is associated with the R-algebraized space (X, A),53 and D a non-trivial Lorentzian A-connection on it (ie, R(D ) = 0). In ADG, the pair (E , D ) is generically referred to as a Y-M ?eld, the triplet (E , ρ, D ) as a Lorentz-Yang-Mills (L-Y-M) ?eld, and it has been shown [67, 68, 70] that54 every Lorentzian vector sheaf yields a (non-trivial) L-Y-M ?eld (E , ρ, D ) on X the (non-vanishing) ?eld strength of which is F (D ). As in the de?nition of the E-L space earlier, in case the curvature F of the connection D of a L-Y-M ?eld satis?es the free Y-M equations, which we write as follows552 2 δE ndE (F ) = 0 or ?E ndE (F ) = 050

(56)

In the closing section we will return to comment thoroughly, in the light of ADG, on this remark by Feynman and the similar one of Isham also quoted in the beginning of the paper. 51 More precisely, in Bergmann’s theoretical scenario for classical Lorentzian gravity, g?ν is replaced by a ?eld of four 2 × 2 Pauli spin-matrices which is locally invariant when conjugated by a member of SL(2, C)—the double cover of the Lorentz group. 52 Later in the present section we will discuss brie?y self-dual connections from ADG’s point of view. 53 With X a paracompact Hausdor? topological space and A a ?ne unital commutative algebra sheaf (over R) on it, as usual. 54 In the sequel, and similarly to how we used di?erent symbols for the (vacuum) gravitational connection D and its Y-M counterpart D, we will use F for the curvature of the latter instead of R (R and R) that we used for the former. In the Y-M context the curvature of a connection is usually referred to as the (gauge) ?eld strength. 55 In (56), ‘δ ’ is the coderivative [46] and ? the Laplacian operator, which we will de?ne in an ADG-theoretic manner shortly. These are two equivalent expressions of the free Y-M equations. Their equivalence, which is a consequence of the covariant di?erential Bianchi identity (50), has been shown in [67].

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and which, in turn, we assume that can be obtained from the variation of a corresponding Y-M action functional YM,56 the curvature space (A, ?, ?1 , d, ?2 ) associated with the L-Y-M ?eld is called a L-Y-M curvature space,57 while the supporting X , a L-Y-M space.58 In connection with the said derivation of the Y-M equations from YM, we note that59 the solutions of the Y-M equations that correspond to a given Y-M ?eld (E , D ) are precisely the critical or stationary points (or extrema) of YM that can be associated with E . In order to make sense of (56) ADG-theoretically, we need to de?ne the coderivative and the Laplacian of a given L-Y-M ?eld (E , ρ, D ). We do this below. 3.2.1 The adjoint δ and the Laplacian ? of an A-connection in ADG

Let T = (A, ?, ?1 ) be the di?erential triad we are working with and ρ a Lorentzian A-metric on it, as usual. Let also E be a Lorentzian vector sheaf of ?nite rank n and D a Lorentzian Y-M connection on it. By emulating the classical situation sheaf-theoretically, as it is customary in ADG, one can de?ne the adjoint derivation δ of D relative to ρ as the following A-morphism of the vector sheaves involved δ 1 ≡ δ : ?1 (E ) ?→ E (≡ ?0 (E )) satisfying ρ(D (s), t) = ρ(s, δ (t)) (58) (57)

We will discuss this derivation in more detail shortly. A particular kind of Bianchi space de?ned earlier. 58 In order for the reader not to be misled by our terminology, it must be noted here that, in contrast to the usual term ‘(free) Yang-Mills ?eld’ by which one understands the ?eld strength of a gauge potential which is a solution to the (free) Y-M equations (56), in ADG, admittedly with a certain abuse of language, a Y-M ?eld is just the pair (E , D), without necessarily implying that F (D) satis?es (56). On the other hand, the Y-M space X supporting the Y-M curvature space (A, ?, ?1 , d, ?2 ) associated with a Y-M ?eld (E , D), is supposed to refer directly to solutions F (D) of (56)—as it were, it represents the ‘solution space’ of (56). This is in complete analogy to the Einstein-Lorentz space and Einstein space X de?ned in connection with the vacuum Einstein equations for Lorentzian gravity in (53). We will return to comment further on this conception of a curvature space as a geometrical ‘solution space’ in section 5 when we express (53) in ?nitary terms. 59 In fact, the statement that follows is a theorem in ADG [68, 70, 71]. We will return to it in 3.3. 60 Which can be de?ned in complete analogy to (58).

Now that we have established with the help of ADG the close structural similarity between vacuum Einstein Lorentzian gravity and free Y-M theory, we will elaborate for a while on our remark earlier that both (53) and (56) or (64) derive from the extremization of an action functional—the E-H EH in the ?rst case, and the Y-M YM in the second. Since only vacuum Einstein gravityAlways remembering that the ?eld strength F of the L-Y-M connection D is an A-morphism between the A-modules E and ?2 (E ) (ie, a member of HomA (E , ?2 (E ))(X )), as (37) depicts.61

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interests us here, we will discuss only the variation of EH, leaving the variation of YM for the reader to read from [67, 68, 71]. As it has been transparent in the foregoing presentation, from the ADG-theoretic point of view, the main dynamical variable in vacuum Einstein Lorentzian gravity is the spin-Lorentzian A-connection D, or equivalently, its gauge potential part A on the vector sheaf E ↑ = (E , ρ). Thus, one naturally anticipates that the E-H action EH is a functional on the a?ne space AA (E ↑) of Lorentzian metric (ie, ρ-compatible) A-connections on E ↑ . Indeed, we de?ne EH as the following map EH : AA (E ↑ ) ?→ A(X ) reading ‘point-wise’ D → EH(D) := R(D) =: tr R(D) (66) (65)

where. plainly, R is a global section of the structure sheaf of coe?cients A (ie, R ∈ A(X )). Our main contention (in fact, a theorem in ADG [67, 68, 70]) in 2.3, as well as in 3.2 in connection with Y-M theory, was that the solutions of the vacuum Einstein ?eld equations (53) that correspond to a given E-L ?eld (E ↑ , D) are obtained from extremizing EH—that is, they are the critical or stationary points of the functional EH associated with E ↑ in (65) and (66) above. In what follows we will recall brie?y how ADG deals with this statement. The critical points of EH can be obtained by ?rst restricting it on a curve γ (t) in connection space (ie, γ : t ∈ R ?→ γ (t) ∈ AA (E ↑ )) and then by in?nitesimally varying it around its ‘initial’ value EH[D0 ] ≡ EH[γ (0)]. Alternatively, and following the rationale in [70], in order to ?nd the stationary points of EH, one has to ?nd the ‘tangent vector’ at time t = 0 to a path γ (t) in the a?ne space AA (E ↑ ) of A-connections of E ↑, on which path EH is constrained to take values in A(X ) as (65) dictates. All in all, one must evaluate EH(γ (t)) (0) ≡EH(γ ) (0) (67)

By setting EH(Dt ) (0) in (70) equal to zero, one arrives at the vacuum Einstein equations (53) for Lorentzian gravity. 3.3.1 A brief note on the topology of AA (E ↑ )

In the introduction we alluded to the general fact that the space of connections is non-linear (ie, it is not a vector space) with a ‘complicated’ topology. Below we would like to comment brie?y on the issue of the topology of the space AA (E ↑ ) of spin-Lorentzian connections on E ↑ . This issue d is of relevance here since one would like to make sense of the dt -di?erentiation of EH in (67). Thus, in connection with (67), the crucial question appears to be: with respect to what topology (on AA (E ↑)) does one take the limit so as to de?ne the (‘variational’) derivative of EH with respect to t (ie, with respect to D) in (67)?63 ADG answers this question by ?rst translating it to an equivalent question about convergence in the structure sheaf A. That is to say, can one de?ne limits and convergence in the sheaf A of coe?cients? To see that this translation is e?ective, one should realize that in order to de?ne the derivative of EH one need only be able to take limits and study convergence in the space where the latter takes values, which, according to (65), is A(X )! Thus, ADG has given so far the following two answers to the question when EH(γ ) is well de?ned: 1. When A is a topological algebra sheaf [67, 68, 70, 71]. 2. When A is Rosinger’s algebra of generalized functions [70, 71]. For in both cases A has a well de?ned topology and the related notion of convergence. In section 5, where we give a ?nitary, causal and quantal version of the vacuum Einstein equations for Lorentzian gravity (53)—them too derived from a variation of a reticular E-HWe will return to de?ne ? shortly. This question would also be of relevance if for instance one asked whether the map (path) γ in (68) is continuous.63 62

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? → action functional EHi , we will give a third example of algebra sheaves—the ?nsheaves of incidence algebras—in which the notions of convergence, limits and topology (the so-called Rota topology) ? → are well de?ned so as to ‘justify’ the corresponding di?erentiation (variation) EH i . The discussion above prompts us to make the following clari?cation: to ‘justify’ the derivation of Einstein’s equations from varying EH with respect to D, one need not study the topology of AA (E ↑ ) per se. Rather, all that one has to secure is that there is a well de?ned notion of (local) convergence in A.64 This is how ADG essentially evades the problem of dealing directly with the ‘complicated’ topology of AA (E ↑). We conclude this discussion of the E-H action functional EH and its variation yielding the vacuum gravitational equations, by giving a concise ADG-theoretic statement about the (gauge) invariance of the ?rst which in turn amounts to the (gauge) covariance of the second. Let E ↑ = (E , ρ) be our usual (real) E-L vector sheaf (of rank 4) and D a spin-Lorentzian gravitational metric connection on it whose curvature R is involved in EH(D) above. Then, the Einstein-Hilbert functional EH is invariant under the action of a (local) ρ-preserving gauge transformation, by which we mean a (local) element (ie, local section) of the structure group sheaf AutA E ↑ ≡ L+ := Autρ E of E ↑ = (E , ρ), which, in turn, is a subsheaf of AutAE , where locally, AutA E (U ) = GL(4, A(U )) = GL(4, A)(U ). 3.3.2 A brief note on ? , the Hodge-? operator, and on self-duality in ADG

Below, we discuss brie?y ` a la ADG the volume element or measure ? appearing in the E-H action integral (69), as well as the Hodge-? operator and the self-dual Lorentzian connections A+ associated with it, thus prepare the ground for a brief comparison we are going to make subsequently between our locally ?nite, causal and quantal vacuum Einstein gravity and an approach to non-perturbative canonical quantum gravity based on Ashtekar’s new variables [3].This is another example of the general working philosophy of ADG according to which the underlying space or ‘domain’ so to speak (here AA (E ↑ )) is of secondary importance for studying ‘di?erentiability’. For the latter, what is of primary importance is the algebraic structure of the objects that live on that domain. For the notion of derivative, and di?erentiability in general, one should care more about the structure of the ‘target space’ or ‘range’ (here the structure sheaf space A) than that of the ‘source space’ or ‘domain’ (here the base space X )—after all, the generic base ‘localization’ space X employed by ADG is assumed to be just a topological space without having been assigned a priori any sort of di?erential structure whatsoever. Of course, in the classical case, X is ∞ completely characterized, as a di?erential manifold, by the corresponding structure sheaf AX ≡ CX of in?nitely di?erentiable (smooth) functions (in particular, see our comments on Gel’fand duality in 5.5.1). In other words, the classical di?erential geometric notions ‘di?erential (ie, C ∞ -smooth) manifold’ and ‘the topological algebra C ∞ (X )’ are tautosemous (ie, semantically equivalent) notions. Alas, other more general kinds of di?erentiability, may come from algebraic structures A other than C ∞ (X ) that one may localize sheaf-theoretically (as structure sheaves AX ) on an arbitrary topological space X . This is the very essence of ADG and will recur time and again in the sequel.64

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31

1. Volume element. Let (X, A) be our usual K-algebraized space and E a free A-module of ?nite rank n over X , which is locally isomorphic to the ‘standard’ one An . Let also ρ be a strongly non-degenerate (and inde?nite, in our case of interest) metric on E , which makes it a pseudo-Riemannian free A-module of ?nite rank n over X . Then, one considers the sequence ? ≡ (?i )1≤i≤n of global sections of E ? An (ie, ?i ∈ An (X ) = A(X )n )—the so-called Kronecker gauge of An .65 Then, the volume element ? associated with the given A-metric ρ is de?ned to be ? := That is to say, the volume element ? is a nowhere vanishing (because ρ is non-degenerate) global section of the structure sheaf A. Moreover, since (∧n An )? (X ) = (∧n (An )? )(X ) = (det An )? (X ) = A(X ), ? can be viewed as an A(X )-linear morphism on det(An ) and, as such, as a map of A into itself: ? ∈ (E ndA)(X ) = EndA = A(X ). The crux of the argument here is that the de?nition (71) of ? readily applies to the case where X is an Einstein space and (E ↑, ρ) our usual (real) Lorentzian vector sheaf on it. This is so because, as mentioned earlier, E ↑ is a locally free A-module of rank 4, that is, locally (ie, U -wise) in X : E ↑ ? A4 . Hence, the volume element ? appearing in (69) is now an element of A(U ). Of course, since, by de?nition, A is a ?ne sheaf, here too ? can be promoted to a global section of A (? ∈ A(X )). 2. Hodge-?. As with the volume element ? , let (E , ρ) be a pseudo-Riemannian (Lorentzian) free A-module of rank n and recall from (12) the canonical A-isomorphism ρ ? between the ρ ? A-modules E and its dual E ? induced by ρ. That is to say, E ? = E ? ≡ HomA (E , A). We de?ne the following A-isomorphism ? of A-modules ? : ∧p E ? ?→ ∧n?p E ? To give ?’s section-wise action, we need to de?ne ?rst, for any v ∈ ∧n?p E (X ), v := (∧n?p ρ ?)(v ) ∈ ∧n?p E ?(X ) = (∧n?p E (X ))? so that then we can de?ne (?u)(v ) := ? (u ∧ v ) ≡ (u ∧ v ) · ? ∈ A(X ) for u ∈ ∧p E ? (X ) = ∧p E (X )? .In ADG, this appellation for ? is reserved for positive de?nite (Riemannian) metrics ρ [67], but here we extend the nomenclature to include inde?nite metrics as well.65

|ρ|?1 ∧ · · · ∧ ?n ∈ (∧n An )(X ) ≡ (det An )(X ) = A(X )

(71)

(72)

(73)

(74)

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Two things can be mentioned at this point: ?rst, that for the identity or unit global section 1 of A, ?1 = ? , and second, that ? entails an A-isomorphism of the A-module de?ned by the exterior algebra of E ? , ∧E ? , into itself. The latter means, in turn, that ? is an element of AutA (∧E ? ). The map ? of (72) and (74) is the ADG-theoretic version of the usual Hodge-? operator induced by the A-metric ρ. 3. Self-dual Lorentzian connections A+ . Now that we have ? at our disposal, we can de?ne a particular class of Y-M A-connections D + on vector sheaves, the so-called self-dual connections, whose gauge potential parts A+ are coined self-dual gauge ?elds. So, we let (E , ρ, D ) be a L-Y-M ?eld on a L-Y-M space X . The de?nition of D + s pertains to the property that their curvatures, F + := F (D + ), satisfy relative to the Hodge-? duality operator ?F + = F + hence their name self-dual. In view of (75) and the second Bianchi identity (49), we have2 + n·3+1 δE ? D n?2 ?)(F + ) = (?1)1+3n ? D n?2 (F + ) = ndE (F ) = ((?1) 2 + = (?1)1+3n ? DE ndE (F ) = 0

(75)

(76)

the point being that the (?eld strengths F + of the) self-dual connections D + also satisfy the Y-M equations. We will return to self-dual connections in section 5 where we will discuss the close a?nity between our ?nitary, causal and quantal version of vacuum Einstein Lorentzian gravity and a recent approach to non-perturbative quantum gravity which uses Ashtekar’s new (canonical) variables [3].

3.4

Y-M and Gravitational Moduli Space: G -Equivalent Connections

In the present subsection we will give a short account of the ADG-theoretic perspective on moduli spaces of L-Y-M connections, focusing our attention on the corresponding moduli spaces of spinLorentzian (vacuum) gravitational connections that are of special interest to our investigations in this paper. To initiate our presentation, we consider a (real) Lorentzian vector sheaf E ↑ = (E , ρ) and we recall from 2.4 the a?ne space AA (E ) of metric A-connections on it (54). From our discussion of G -sheaves in 2.2, we further suppose that E ↑ is the associated sheaf of the principal sheaf L+ := AutAE ↑ ≡ AutρE —the group sheaf of ρ-preserving A-automorphisms of E (the structure group sheaf of E ↑ , which is also the (local) invariance group of the free Y-M action functional YM(D ) [67, 68]).66 Our main contention in this section is thatIn the case of the functional EH(D) on (E ↑ , D), we saw in the previous subsection that its (local) invariance (structure) group is precisely (AutA E ↑ )(U ) := Γ(U, AutA E ↑ ) ≡ (Autρ E )(U ) =: L+ (U ) ? L↑ .66

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the (global) gauge group AutAE ↑ (X ) ≡ AutA E ↑ ≡ L+ (X ) := Autρ E acts on the a?ne space AA (E ↑ ) of metric A-connections on the Lorentzian vector sheaf E ↑ = (E , ρ). Let us elaborate a bit on the statement above, which will subsequently lead us to de?ne moduli spaces of gauge-equivalent connections. We have already alluded to the fact, in connection with the (local) transformation law of gauge potentials A of A-connections D on general vector sheaves E at the end of 2.1, that one may be g ′ able to establish an equivalence relation A ? A between them, for g a local gauge transformation (ie, a local section of the structure G -sheaf AutA(E ) of E ; g ∈ AutA (E )(U ) = GL(n, A)(U )). We can extend this equivalence relation from the gauge potentials A to their full connections D , as follows. ′ Schematically, and in general, for an A-module E we say that two connections D and D on it are gauge-equivalent if there exists an element g ∈ Aut(E ) making the following diagram commutative E gc

called the orbit of an A-connection D on E under the action α of the gauge group G = AutE on ′ AA (E ). OD consists of all connections D in AA (E ) that are gauge-equivalent to D . Following [67, 68], we would also like to note that it can be shown that the gauge-orbit OD in (83) can be equivalently written in terms of the induced connection DE ndE as follows OD = {D ? DE ndE (g )g ?1 : g ∈ AutE} (84)

which means that the stability group of the connection D ∈ AA (E ) consists of all those (gauge) transformations of E (g ∈ AutE ) that commute with D . At this point, and before we de?ne moduli spaces of gauge-equivalent connections ADGtheoretically, we would like to digress a bit and make a few comments on the possibility of developing di?erential geometric ideas (albeit, not of a classical, geometrical C ∞ -smooth sort, but of an algebraic ADG kind) on the a?ne space AA (E ). The remarks below are expressed in order to prepare the reader for comments on the possibility of developing di?erential geometry on the gauge moduli space of gravitational connections that we are going to make in 5.3 in connection with some problems (eg, Gribov’s ambiguity) people have encountered in trying to quantize general relativity (regarded as a gauge theory) both canonically (ie, in a Hamiltonian fashion) and covariantly (ie, in a Lagrangian fashion). It is exactly due to these problems that others have also similarly felt the need of developing di?erential geometric concepts and constructions (albeit, of the classical, C ∞ -sort) on moduli spaces of Y-M and gravitational connections [7, 8]. As a ?rst di?erential geometric idea on AA (E ), we would ?rst like to de?ne a set of objects (to be regarded as abstract ‘tangent vectors’) that would qualify as the ‘tangent space’ of AA (E ) at any of its points D , and then, after we de?ne moduli spaces of gauge-equivalent connections below, we would also like to de?ne an analogous ‘tangent space’ to the moduli space at a gauge-orbit OD of a connection D ∈ AA (E ). We saw earlier (2.4) that for E a vector sheaf of rank n, the a?ne space AA (E ) can be modelled after ?1 (E ndE )(X ). We actually de?ne the latter space to be the sought after ‘tangent space’ of AA (E ) at any of its ‘points’ D . That is to say, T (AA (E ), D ) := ?1 (E ndE )(X ) (86)

As a matter of fact, one can actually prove (86) along classical lines—for example, by ?xing a point D in

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35

We are now in a position to de?ne the global moduli space or gauge orbit space of the Aconnections on E , as follows M (E ) ≡ AA (E )/AutE := OD =D

OD

(87)

D∈AA (E )

The epithet ‘global’ above indicates that the quotient in (87) can be actuallly localized—something that comes in handy when one, as we do, works with a vector sheaf E on X and the latter is gauged relative to a local frame U = {U }. The localization of M (E ) means essentially that one uses the sheaf of germs of moduli spaces of the A-connections of the module or vector sheaf E in focus. To see this, the reader must realize that, as U ranges over the open subsets of X , one deals with a (complete) presheaf of orbit spaces equipped with the obvious restriction maps. To follow this line of thought, one ?rst observes the inclusion AA (E )|U ? AA|U (EU ) (88)

for every open U in X . So, in complete analogy to (81), one has the action of Aut(E|U ) on the local sets AA (E )|U of A-connections in (88) Aut(E|U ) × AA (E )|U ?→ AA (E )|U entailing the following ‘orbifold sheaf’ of gauge-equivalent A-connections on E M(E ) = AA (E )/AutE M(E ) is the aforesaid sheaf of germs of moduli spaces of A-connections on E .the a?ne space AA (E ), regard it as ‘origin’ (ie, the zero vector 0), let a curve γ (t) in AA (E ) pass through it (ie, D ≡ D0 = γ (0)), and then ?nd the vector γ ˙ (t) tangent to γ . This proof has been shown to work in the particular case the structure sheaf A is a topological vector space sheaf [68, 71] (and in section 5 we will see that it also works in the case of our ?nsheaves of incidence algebras for deriving the locally ?nite, causal and quantal vacuum Einstein equations for Lorentzian gravity); in fact, we used it in (67) and (68) to derive the vacuum Einstein equations from a variational principle on the space of Lorentzian connections.

(91)

(92)

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Anastasios Mallios and Ioannis Raptis

Finally, it must also be mentioned here, in connection with the local isomorphism E ? An of a vector sheaf E mentioned earlier, that (AutA E )(U ) above reduces locally to GL(n, A)(U ) = GL(n, A(U )), as follows68 (AutA E )(U ) = Aut(E|U ) = Aut(An |U ) = (AutAn )(U ) = ? = Mn (A) (U ) ≡ GL(n, A)(U ) ≡ GL(n, A(U )) We can distill this to the following remark any local automorphism of a given vector sheaf E of rank n over one of its local gauges U is e?ectively given by a local automorphism of An —that is to say, by an element of GL(n, A(U )) = GL(n, A)(U ) ≡ GL(n, A|U ). so that the gauge (structure) group AutA E of E is locally (ie, U -wise) reduced to the group sheaf GL(n, A),69 as it has been already anticipated, for example, in 2.1.2 in connection with the transformation law of gauge potentials,70 and earlier in connection with vacuum Einstein Lorentzian gravity on E ↑ . As noted before, now that we have de?ned moduli spaces of gauge-equivalent connections, and similarly to the ‘tangent space’ T (AA (E ), D ) in (86), we would like to de?ne T (OD , D )—the ‘tangent space’ to a gauge-orbit of an element D ∈ AA (E ) and, in extenso, T (M (E ), OD )—the ‘tangent space’ to the moduli space of E at an orbit of D ∈ AA (E ). We have seen how the induced A-connection of the vector sheaf E ndE DE ndE : E ndE ?→ ?1 (E ndE ) (94) (93)

for u ∈ ?1 (E ndE )(X ). Of course, for u = 0 ∈ ?1 (E ndE )(X ), one sees that D belongs to SD , so that SD is a subspace of AA (E ) through D . In fact, one can show [68, 71] that SD is an 1 71 a?ne C-linear subspace of AA (E ) through the point D , modelled after (kerδE ndE )(X ).68 69

In the case of E ↑ , the local reduction below has already been anticipated earlier. Or equivalently, to its complete presheaf of sections Γ(GL(n, A)). 70 See remarks after (9). 71 1 1 (kerδE ndE )(X ) being in fact a sub-A(X )-module of ? (E ndE )(X ).

for any local gauge U of E . 1 In toto, since both DE ndE and δE ndE are restricted on the gauge group AutE , and in view of (84), one realizes that1 ⊥ T (OD , D ) = im(DE ndE|AutE ) = ker(δE ndE|AutE )

Now that we have M (E ) we are in a position to de?ne similarly moduli spaces of (self-dual) spin-Lorentzian connections. Of course, our de?nition of ‘tangent spaces’ on OD and on M (E ) above carries through, virtually unaltered, to the particular (self-dual) Lorentzian case. As noted above, this will become relevant in section 5 where, in view of certain problems that both the canonical and the covariant quantization approaches to quantum general relativity (based on the Ashtekar variables) encounter, the need to develop di?erential geometric ideas and techniques on the moduli space of (self-dual) spin-Lorentzian connections has arisen in the last decade or so.

Finally, in a possible covariant quantization scenario for vacuum Einstein Lorentzian gravity that we are going to discuss in section 5, M(E ↑ ) may be regarded as the (quantum) con?guration space of the theory in a way analogous to the scheme that has been proposed in the context of Ashtekar’s new variables for non-perturbative canonical quantum gravity [3, 6, 7, 8]. In connection with the latter, we note that since the main dynamical variable is a self-dual spin-Lorentzian connection D+73 (see end of 3.3), the corresponding moduli space is denoted by+ ↑ ↑ ↑ M+ (E ↑) = A+ A (E )/AutA E ≡ AA (E )/Autρ E

One of our main main aims in this paper is to show that the general ADG-theoretic concepts and results presented in the last two sections are readily applicable in the particular case of the curved ?nsheaves of qausets perspective on (the kinematics of) Lorentzian gravity that has beenWhich in turn means that (E ↑ , D(+) ) ≡ (E , ρ, D(+) ) de?nes a (self-dual) E-L ?eld. Or again, locally, its gauge potential part A+ . 74 Always remembering of course that L↑ = SO(1, 3)↑ and its double covering spin-group SL(2, C) are only locally (ie, Lie algebra-wise) isomorphic (ie, sl(2, C) ? so(1, 3)↑ ). Also, for a general (real) Lorentzian vector sheaf (E , ρ) of rank n, which locally reduces to An (ie, it is a locally free A-module), its local (structure) group of Lorentz transformations is Autρ E (U ) = SL(n, A)(U ) ≡ SL(n, A(U )) ? AutA E (U ) = GL(n, A)(U ) ≡ GL(n, A(U )) ≡ Mn (A)? (U ) = (E ndA E )? (U ).73 72

Finitary, Causal and Quantal Vacuum Einstein Gravity

39

developed in the two past papers [74, 75]. In the present section, we recall in some detail from [74], always under the prism of ADG, the main kinematical structures used for a locally ?nite, causal and quantal version of vacuum Einstein Lorentzian gravity, thus we prepare the ground for the dynamical equations to be described ‘?nitarily’ in the next. In the last subsection (4.3), and with the reader in mind, we give a concise r? esum? e—a ‘causal ?nitarity’ manual so to speak—of some (mostly new) key kinematical concepts and constructions to be described en passant below. More analytically, we will go as far as to present a ?nitary version of the (self-dual) moduli space M(+) (E ↑) in (103) and (104) above—arguably, the appropriate (quantum) kinematical con?guration space for a possible (quantum) theoresis of the (self-dual) spin-Lorentzian connec(+) tions Ai inhabiting the aforesaid ?nsheaves of qausets. We will also present, based on recent results about projective and inductive limits in the category DT of Mallios’ di?erential triads [81, 82], as well as on results about projective limits of inverse systems of principal sheaves endowed with Mallios’ A-connections [124, 125, 126], the recovery, at the projective limit of in?nite re?nement (or localization) of an inverse system of principal ?nsheaves of qausets and reticular spin-Lorentzian connections on them, of a structure that, from the ADG-theoretic perspective, comes very close to, but does not reproduce exactly, the kinematical structure of classical gravity in its gauge-theoretic guise—the principal orthochronous Lorentzian ?ber bundle P ↑ over a C ∞ -smooth spacetime manifold M endowed with a non-trivial (self-dual) smooth spin-Lorentzian connection D(+) on it (subsection 4.2).75 In this way, we are going to be able to make brief comparisons, even if just preliminarily at this early stage of the development of our theory, between a similar di?erential geometric scheme on the moduli space of gauge-equivalent spin-Lorentzian connections that has been worked out in [8], like ADG, through entirely algebraic methods.76 However, and this must be stressed from the start, unlike [8], where projective limit techniques are used in order to endow (a completion of) the moduli space of gauge-equivalent connections with a di?erential manifoldlike structure, thus (be able to) induce to it classical di?erential geometric notions such as di?erential forms, exterior derivatives, vector ?elds, volume forms etc, we, with the help of ADG, already possess those at the ?nitistic and quantal level of the curved ?nsheaves of qausets. Moreover, our projective limit result—the smooth differential triad, Lorentzian principal sheaf and non-trivial connection on it which, as noted above, closely resembles the classical C ∞ -diferential triad as well as the principal orthochronous Lorentz sheaf (bundle) and its associated curved locally Minkowskian vector sheaf (bundle) over the C ∞ -smooth manifold M of general relativity—only illustrates the ability of our discrete algebraic (quantal) structures to yield at the (corThe word ‘emulates’ above pertains to the fact that our projective limit triad (as well as the principal sheaf and spin-Lorentzian connection relative to it) will be seen not to correspond precisely to the classical di?erential ∞ triad (AX ≡ K CX , ?, ?1 ), but to one that in the context of the present ADG based paper may be regarded as a ‘generalized smooth’ triad (write smooth for short). This smooth triad’s structure sheaf will be symbolized by K ∞ AX ≡ K C∞ X in order to distinguish it from the CX employed in the classical case. On the other hand, we will be using the same symbols for the ?at 0-th order nilpotent derivation d0 ≡ ? as well as the A-module of 1st order di?erential forms ?1 in the C∞ -smooth and the usual C ∞ -smooth triads. 76 For, to recall Grauert and Remmert: “The methods of sheaf theory are algebraic.” [47]. The purely algebraic character of ADG has been repeatedly emphasized in [67, 68, 70, 77, 78, 74, 75, 73, 71].75

40

Anastasios Mallios and Ioannis Raptis

respondence) limit of in?nite localization or re?nement of the qausets a structure that emulates well the kinematical structure of classical Lorentzian gravity [92, 74, 93, 75]. At the same time, and perhaps more importantly, this indicates, in contrast to [8] where projective limits are employed in order to produce ‘like from like’ (ie, induce a classical di?erential geometric structure from inverse systems of di?erential manifolds), what we have repeatedly stressed here, namely that, to do di?erential geometry—the di?erential geometric machinery so to speak—is not inextricably tied to the C ∞ -smooth manifold, so that we do not depend on the latter to provide us with the standard, and by no means unique, necessary or ‘preferred’, di?erential mechanism usually supplied by the algebra C ∞ (M ) of smooth functions on the di?erential manifold M as in the classical case. Our di?erential geometric machinery, as we shall see in the sequel, comes straight from the (incidence) algebras inhabiting the stalks of vector, di?erential module and algebra sheaves like the generic locally free A-modules E of ADG above, over a ?nitary topological base space(time) without mentioning at all any di?erential structure that this base space should a priori be equipped with, and certainly not the classical C ∞ -manifold one. In other words, our di?erential geomet∞ ric machinery does not come from assuming CM as structure sheaf in our ?nitary, 77 ADG-based constructions. We would like to distill this to the following slogan that time and again we will encounter in the sequel: Slogan 1. Di?erentiability derives from (algebras in) the stalk (in point of fact, from the structure sheaf A of coe?cients or generalized arithmetics), not from the base space.78 Then, the upshot of our approach to all the structures to be involved in the sequel is that in the spirit of ADG [67, 68, 70, 74, 75, 77, 78] and what has been presented so far here along those lines, everything to be constructed below, whether kinematical or dynamical, is manifestly independent of a background C ∞ -smooth spacetime manifold M , its ‘structure group’ Di?(M ) and, as a result, of the usual di?erential geometry (ie, calculus) that such a base space supports. In a nutshell, our (di?erential) geometric constructions are genuinely background C ∞ -manifold free.

Interestingly enough, such a position recurs time and again, as a leit motiv so to speak, in the Ashtekar quantum gravity program [4, 5]. But let us now go on to more details.A similar point was made in footnotes 8 and 64, for example. We will return to discuss it in more detail in the concluding section. 78 As we have said many times, the classical case corresponding to taking for base space X (a region of) the ∞ smooth manifold M and for AX its structure sheaf CX —the sheaf of germs of sections of in?nitely di?erentiable functions on X .77

Finitary, Causal and Quantal Vacuum Einstein Gravity

41

4.1

Principal Finsheaves and their Associated Finsheaves of Qausets

First, we give a short account of the evolution of our ideas leading to [74] and [75] which the present paper is supposed to continue as it takes a step further into the dynamical realm of qausets.79 4.1.1 A brief history of ?nitary spacetime and gravity

Our entire project of developing a ?nitary, causal and quantal picture of spacetime and gravity started with Sorkin’s work on discrete approximations of continuous spacetime topology [114]. Brie?y, Sorkin showed that when one substitutes the point-events of a bounded region X of a topological (ie, C 0 ) spacetime manifold M by ‘coarse’ regions (ie, open sets) U about them belonging to a locally ?nite open cover Ui of X , one can e?ectively replace the latter by locally ?nite partially ordered sets (posets) Pi which are T0 -topological spaces in their own right and, e?ectively, topologically equivalent to X . Then, these posets were seen to constitute inverse ← ? systems P = (Pi , ) of ?nitary topological spaces, with the relation Pj Pi being interpreted as ‘the act of topological re?nement or resolution of Pi to Pj ’.80 Sorkin was also able to show, under reasonable assumptions about X ,81 that the Pi s are indeed legitimate substitutes of it in that at the inverse or projective limit of in?nite re?nement, resolution or localization of the Ui s and their associated Pi s, one recovers the C 0 -region X (up to homeomorphism). Formally one writes ← ? lim Pi ≡ P∞ lim ← ? P ≡ ∞← ihomeo.

? X

(105)

Subsequently, by exploring ideas related to Gel’fand duality,82 which had already been anticipated in [129], Raptis and Zapatrin showed how to associate a ?nite dimensional, associative and ← ? noncommutative incidence Rota K-algebra ?i with every Pi in P , and how these algebras can be interpreted as discrete and quantum topological spaces bearing a non-standard topology, called the Rota topology, on their primitive spectra83 [92]. They also showed, in a way reminiscent of ˇ the Alexandrov-Cech construction of nerves associated with locally ?nite open covers of manifolds, how the Pi s may be also viewed as simplicial complexes84 as well as, again by exploringFor a more detailed and thorough description of the conceptual history of our work, as well as of its relation with category and topos theory, the reader is referred to the recent work [90]. A topos-theoretic treatment of ?nitary, causal and quantal Lorentzian gravity is currently under way [91]. 80 Meaning essentially that the open covering Ui of X from which Pi derives is a subcover of (ie, coarser than) Uj . Roughly, the latter contains more and ‘smaller’ open sets about X ’s points than the former. In this sense, acts of ‘re?nement’, ‘resolution’, or ‘localization’ are all synonymous notions. That is, one re?nes the coarse open sets about X ’s point-events and in the process she localizes them (ie, she e?ectively determines their locus) at higher resolution or ‘accuracy’. As be?ts this picture, Sorkin explicitly assumes that “the points of X are the carriers of its topology” [114]. 81 For instance, X was assumed to be relatively compact (open and bounded) and (at least) T1 . 82 We will comment further on Gel’fand duality in the next section. 83 That is, the sets of the incidence algebras’ primitive ideals which, in turn, are kernels of irreducible representations of the ?i s. 84 See also [130, 128] about this.79

42

Anastasios Mallios and Ioannis Raptis

a variant of Gel’fand duality, how there is a contravariant functor between the category P of ?nitary substitutes Pi and poset morphisms85 between them, and the category Z of the incidence algebras ?i associated with the Pi s and injective algebra homomorphisms between them. Below, we would like to highlight three issues from the investigations in [92]: 1. Since the ?i s are objects dual to the Pi s which, in turn, are discrete homological objects (ie, ?nitary simplicial complexes) as mentioned above, they (ie, the incidence algebras) can be viewed as discrete di?erential manifolds [26, 28, 27, 128]. Indeed, they were seen to be reticular spacesAi Di

2. Since now the ?i s are seen to be structures encoding not only topological, but also di?erential geometric information, it was intuited that an inverse—or more accurately, since the incidence algebras are objects Gel’fand-dual to Sorkin’s topological posets—a direct system ? → R = {?i } of the ?i s should yield, now at the direct or inductive limit of in?nite re?nement of the Ui s as in (105), an algebra ?∞ whose commutative subalgebra part A∞ corresponds to (K) C ∞ (X )—the algebra of (K = R, C-valued) smooth coordinates of the point-events of (K) ∞ X , while ?p C (X )-bimodules of smooth di?erential p-forms cotangent at ∞ in D∞ to the each and every point-event of X which, in turn, can now be regarded as being a smooth region of a C ∞ -manifold M .87 We will return to discuss further this limit in subsection 4.2.Monotone maps continuous in the topology of the Pi s. In (106), Ai ≡ ?0 i is a commutative subalgebra of ?i called the algebra of coordinate functions in ?i , while p≥1 p Di ≡ ? a linear subspace of ?i called the module of di?erentials over Ai . The elements of each linear i i subspace ?p of ? in D were seen to be discrete analogues of (exterior) di?erential p-forms. We also note that in i i i the sequel we will use the same boldface symbol ‘Ai ’ and ‘Di ’ to denote the algebra of reticular coordinates and the module of discrete exterior di?erentials over it as well as the ?nsheaves thereof. 87 In retrospect, and as we shall see in the sequel from an ADG-theoretic perspective, that initial anticipation in [92, 93]—that is, that at the inductive limit of in?nite localization of the ?i s one should recover the classical smooth structure of a C ∞ -manifold—was wrong, or better, slightly misled by the classical C ∞ -theory. In fact, as noted earlier, based on ADG results about inverse and direct limits of di?erential triads, we will argue subsequently that at the continuum limit one recovers a smooth algebra structure K C∞ (X ) and K C∞ (X )-bimodules ?p ∞ of smooth p-forms over it, and that both of which may be regarded as ‘generalized’, albeit close, relatives of the corresponding classical C ∞ -ones. Thus, rather than directly anticipate that one should obtain the local smooth structure of a C ∞ -manifold at the inductive limit of in?nite re?nement (of the incidence algebras), perhaps it is more correct at this point just to emphasize that passing from the poset to the incidence algebraic regime one catches a glimpse not only of the topological, but also of the di?erential structure of discretized spacetime. This essentially shows that the di?erential operator—the heart and soul of di?erential geometry—comes straight from the algebraic structure. Equivalently, incidence algebras provide us with a (reticular) di?erential geometric mechanism, something that the ‘purely topological’ ?nitary posets were unable to supply since they are merely86 85

Finitary, Causal and Quantal Vacuum Einstein Gravity

43

3. The aforesaid continuum limit was physically interpreted as Bohr’s correspondence principle, in the following sense: the local (di?erential) structure of classical C ∞ -smooth spacetime should emerge at the physically ‘ideal’ (or operationally ‘non-pragmatic’) limit of in?nite localization of the alocal, discrete and quantal algebraic substrata ?i .88 In the sequel, following Sorkin’s dramatic change of physical interpretation of the locally ?nite posets Pi in [114] from ?nitary topological spaces to causal sets (causets) Pi [18],89 the corresponding reticular and quantal topological spaces ?i where similarly interpreted as quantum causal sets (qausets) ?i [86].90 Qausets, like their causet counterparts, were regarded as locally ?nite, causal and quantal substrata underlying the classical Lorentzian spacetime manifold of macroscopic gravity.91 On the other hand, it was realized rather early, almost ever since their inception in [18, 116], that causets are sound models of the kinematical structure of (Lorentzian) spacetime in the quantum deep, so that in order to address genuinely dynamical issues vis-` a-vis quantum gravity, causet theory should also suggest a dynamics for causets. Thus, how can one vary a locally ?nite poset? has become the main question in the quest for a dynamics for causets92 [90]. It was roughly at that point, when the need to develop a dynamics for causets arose, that ADG entered the picture. In a nutshell, we intuited that a possible, rather general answer to the question above, is by sheaf-theoretic means! in the sense that the fundamentally algebraic methods of sheaf theory, as employed by ADG, could be somehow used to model a realm of dynamically varying causets or, preferably, due to a quantum theoresis of (local) causality and gravity that we had in mind, of their qauset descendants.associative multiplication structures (ie, arrow semigroups, or monoids, or even poset categories) and not linear structures (ie, one is not able to form di?erences of elements in them). This remark will be of crucial importance subsequently when we will apply ADG-theoretic ideas to these discrete di?erential algebras. 88 For further remarks on this limiting procedure and its physical interpretation, the reader is referred to [92, 74, 93, 75, 131]. We will return to it in an ADG-theoretic context in the next subsection where, as noted above, we will show that one does not actually get the classical C ∞ -smooth structure at the continuum limit, but a C∞ -smooth one akin to it. We will also argue that this (ie, that we do not get back the C ∞ -smooth spacetime manifold at the projective/inductive limit of our ?nitary structures) is actually welcome when viewed from the ADG perspective of the present paper. 89 For a thorough account of this semantic switch from posets as discrete topologies to posets as locally ?nite causal spaces, the reader is referred to [115]. 90 The reader should note that, in accordance with our convention in [86, 74, 75], from now on all our constructions referring to reticular causal structures like the Pi s and their associated ?i s, will bear a right-pointing arrow over them just to remind us of their causal interpretation. (Such causal arrows should not be confused with the right-pointing arrows over inductive systems.) 91 That causality, as a partial order, determines not only the topology and di?erential structure of the spacetime manifold as alluded to above, but also its conformal Lorentzian metric structure of (absolute) signature 2, has been repeatedly emphasized in [18, 113, 116, 117]. 92 Rafael Sorkin in private correspondence.

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Anastasios Mallios and Ioannis Raptis

However, in order to apply the concrete sheaf-theoretic ideas and techniques of ADG to qausets, it was strongly felt that we should somehow marry ?rst Sorkin’s original ?nitary posets in [114] with sheaves proper. Thus, ?nitary spacetime sheaves (?nsheaves) were de?ned as spaces Si of (algebras of) continuous functions on Sorkin’s T0 -posets Pi that were seen to be locally homeomorphic to each other [87].93 The de?nition of ?nsheaves can be captured by the following commutative diagram which we borrow directly from [87]

X σ ≡ π ?10 CX

fi E Pi?1 πi ≡ σi

c

? f i

E

c

SP i

(107)

0 ? where CX is the usual sheaf of germs of continuous functions on X , while fi and f i are continuous 0 surjections from the topological spaces X and CX to the ?nitary topological spaces Pi and Si , respectively.

Now, the diagram (107) above prompts us to mention that the complete analogy between Sorkin’s ?nitary topological posets Pi and ?nsheaves Si rests on the result that an inverse system ← ? S = (Si , ?) of the latter was seen in [87] to possess a projective limit sheaf S∞ ≡ SP∞ 94 that is 0 homeomorphic to CX —the sheaf of germs of sections of continuous functions on the topological spacetime manifold X . That is to say, similarly to (105), one formally writes,

← ? lim Si ≡ S∞ lim ← ? S ≡ ∞← i

homeo.

0 ? CX

(108)

One could cast the result above as a limit of commutative diagrams like the one in (107) which

93

That is, one formally writes Pi ? U ? Si (U ), where πi is the continuous projection map from the sheaf spaceπi

σi

Si to the base topological poset Pi , σi its inverse (continuous local section) map and U an open subset of Pi . In ?1 other words, for every open U in Pi : πi ? σi (U ) = U ? [?U ∈ Pi : σi = πi ] (ie, σi is a local homeomorphism having πi for inverse) [87, 67]. Here we symbolize these ?nsheaves by Si ≡ SPi .94

From (105), P∞

homeo.

?

X.

Finitary, Causal and Quantal Vacuum Einstein Gravity

45

de?nes ?nsheaves, as follows Pi ? ? . . . ? ?i ? ? ? →

π ?1 σi

fij

ij ?1 πj

? ij

Pj

? ? ? →σj

Si ? ?? . . . ? ??

fij

Sj

(109)

fj ∞ ?fij =:fi∞

i∞

? i∞ 0 ? X ? ? ? → CX σ π ?1 homeo.

? ? fi∞ :=f j ∞ ?fij

∞←i

lim Pi ≡ P∞

homeo.

? S∞ ≡ lim Si∞←i

? with fij and f ij continuous injections—the ‘re?nement’ or ‘localization arrows’—between the Pi s ← ? ← ? in P and the Si s in S , respectively.95 Having ?nsheaves in hand, our next goal was to materialize ADG-theoretically our general answer to Sorkin’s question mentioned above. The basic idea was the following: Since sheaves of (algebraic) objects of any kind may be regarded as universes of variable objects [67, 64], by (sheaf-theoretically) localizing or ‘gauging’ the incidence Rota algebras modelling qausets over the ?nitary topological posets Pi or their locally ?nite causet descendants Pi ,96 the resulting ?nsheaves would stand for worlds of variable qausets—ones varying dynamically under the in?uence of a locally ?nite, causal and quantal version of gravity in vacuo which, in turn, could be concisely encoded in non-?at connections on those ?nsheaves [74]. Moreover, and this cannot be overemphasized here, by using the rather universal sheaf-theoretic constructionsThese arrows capture precisely the partial order (or net) re?nement relations and ? between the ?nitary ← ? ← ? fij posets in P and their corresponding ?nsheaves in S respectively, as (109) depicts (eg, we formally write: Pi ?→ Pj ≡ Pj ij Pi ). Also from (109), one notices what we said earlier in connection with (105) and (108), namely, 0 that X and CX are obtained at the categorical limit of in?nite (topological) re?nement or localization ( i∞ and ? i∞ ) of the Pi s and the Si s, respectively. 96 For instance, one could regard Pi as a topological space proper by assigning a ‘causal topology’ to it, as for example, by basing such a topology on ‘open’ sets of the following kind: I ? (x) := {y ∈ Pi : y → x} (?x ∈ Pi ) (‘lower’ or ‘past-set topology’), or dually on: I + (x) := {y ∈ Pi : x → y } (‘upper’ or ‘future-set topology’), or even on a combination of both—ie, on ‘open’ causal intervals of the following sort: A(x, y ) := I + (x) ∩ I + (y ) (the so-called Alexandro? topology). It is one of the basic assumptions about the causets of Sorkin et al. that the cardinality of the Alexandro? sets A(x, y ) is ?nite—the so-called local ?niteness property of causets [18]. As basic open sets generating the three topologies above, one could take the so-called covering past, covering future and ? null Alexandro? ‘open’ sets, respectively. These are Ic (x) := {y ∈ Pi : (y → x) ∧ (?z ∈ Pi : y → z → x)}, + Ic (x) := {y ∈ Pi : (x → y ) ∧ (?z ∈ Pi : x → z → y )} and A0 (x, y ) = ? ? (x → y ) ∧ (?z ∈ Pi : x → z → y ), ? respectively. (Note: the immediate arrows in the Hasse diagram of any poset P appearing in the de?nition of Ic , + Ic and An (x, y ) are called covering relations or links and they correspond to the transitive reduction of the partial order based at each vertex in the directed and transitive graph of P . In turn, the three topologies mentioned above can be obtained by taking the transitive closure of these links [86, 75].)95

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Anastasios Mallios and Ioannis Raptis

of ADG, we could carry virtually all the usual C ∞ -di?erential geometric machinery on which the mathematical formulation of general relativity rests, to the locally ?nite setting of ?nsheaves of qausets [75]—the principal di?erential geometric objects being, of course, the aforesaid connections on the relevant ?nsheaves, which implement the dynamics of qausets. Thus, as a ?rst step in this development, we set out to de?ne (curved) principal ?nsheaves ? ? → ? ? → := AutAi ?Pi ≡ Auti ?i of qausets, and their associated ?nsheaves ?Pi ≡ ?i , over a causet Pi .97 By establishing ?nitary versions of the classical general relativistic principles of equivalence and locality, we realized that the (local) structure (gauge) symmetries of ?i are ?nitary correspondents of the orthochronous Lorentz Lie group (ie, locally in Pi one writes for? ? → 98 and that they could thus be organized into the aforesaid mally: AutAi ?Pi (U ) = SO (1, 3)↑ i ), ↑ Gi -?nsheaves Pi . Then, by de?nition, the ?i s are the associated ?nsheaves of the principal Pi↑ s. From the start we also realized that the localization or ‘gauging’ of qausets in the Pi↑ s and their associated ?i s meant that these ?nsheaves could be endowed with non-trivial (ie, non-?at) reticular spin-Lorentzian connections Di ` a la ADG. Indeed, in complete analogy to the general ADG case, after having de?ned reticular ?at connections as the following K-linear and sectionwise Leibniz condition (2)-obeying ?nsheaf morphisms Pi↑0 1 d0 i ≡ ?i : ?i ≡ Ai ?→ ?i

(110)

as in (1), as well as higher order extensionsp p+1 dp , (N ? p ≥ 1) i : ?i ?→ ?i

(111)

between the vector subsheaves ?p i of ?i , we de?ned in [74] non-?at connections Di on the ?nsheaves ?i of ?nite dimensional di?erential Ai -bimodules ?i 99 again as the following K-linear and section-wise Leibniz condition-obeying (4) ?nsheaf morphisms D i : E i ≡ ?? i ?→ Ei ?Ai ?i ≡ ?i (Ei )97

(112)

similarly to (3).100 Moreover, in complete analogy to the local expression for the abstract D s inIn what follows we will be often tempted to use the same epithet, ‘principal’, for both the Pi↑ s and their associated ?i s. We do hope that this slight abuse of language will not confuse the reader. As we will see in the sequel, this identi?cation essentially rests on our assuming a general Kleinian stance towards (physical) geometry whereby ‘states’ (of a physical system) and the ‘symmetry group of transformations of those states’ are regarded as being equivalent. 98 Where U is an open set in Pi regarded as a causal-topological space (see footnote 96 above). 99 The reader should have gathered by now that in the stalks of the structure ?nsheaves Ai dwell the (causal versions Ai of the) abelian (sub)algebras Ai (of ?i ) in (106), while in the ?bers of Di the (causal versions Di of the) Ai -modules Di in (106). 100 The reader should note in connection with (112) that the ‘identi?cation’ Ei ≡ ?? i tacitly assumes that there ?i between them and their is a (Lorentzian) metric ρi on the vector sheaves Ei e?ecting canonical isomorphisms ρ dual di?erential module (covector) ?nsheaves ?i , as in (12). We will give more details about ρi and the implicit

? ? → and the reticular gauge potentials Ai of the Di s above were readily seen to be Auti -valued local ↑ 101 sections of ?1 in analogy with both i (ie, ‘discrete’ so(1, 3)i ? sl(2, C)i -valued local 1-forms), the classical and the abstract (ADG) theory. At this point, we must stress a couple of things about these ?nitary spin-Lorentzian connections Di vis-` a-vis the general ADG theory presented in the previous two sections. 1. About the base space. As it was mentioned in [87], [74] and [75], in our ?nitary regime there are mild relaxations of the two basic conditions of paracompactness and Hausdor?ness (T2 -ness) that ADG places on the base topological space X on which the vector sheaves E bearing connections D are soldered. As noted in footnote 81, the starting region X of the topological spacetime manifold M from which the Pi s (and their associated ?i s) come from was assumed in [114] to be relatively compact and (at least) T1 . If one relaxes paracompactness to relative compactness, and T2 -ness to T1 -ness (and we are indeed able to do so without any loss of generality),102 one is still able to carry out in the locally ?nite regime the entire spectrum of the ADG-theoretic constructions described in the last two sections.103identi?cation of the ?nitary vectors in Ei with their corresponding forms in ?i shortly. For the time being, we note that we would like to call Di ‘the (f )initary, (c)ausal and (q)uantal (v)acuum dynamo’ (fcqv-dynamo) for a reason to be explained in the next section. 101 Of course, since the ?i s are curved, they do not admit global sections [67, 74]. In view of the name ‘fcqvdynamo’ we have given to Di in the previous footnote, its gauge potential part Ai may be ?ttingly coined a ‘fcqv-potential’. The fcqv-potential, like its abstract analogue ω in (6)–(8), is an n × n-matrix of sections of local i 1 reticular 1-forms (ie, Ai ≡ (Ai pq ) ∈ Mn (?i (U )), U open in Pi ). Also, since the local structure of the gauge group Gi of the ?i s is the reticular orthochronous Lorentz Lie algebra so(1, 3)↑ i , we will denote the vector ?nsheaves Ei above as Ei↑ = (Ei , ρi ), in accord with our notation earlier for the (real) orthochronous Lorentzian vector sheaves E ↑ = (E , ρ) of rank 4 in the context of ADG. (However, to avoid uncontrollable proliferation of symbols and eventual typographical congestion of indices, superscripts etc, we will not denote the dual spaces ?i s of the Ei↑ s by ↑ ?↑ i .) Moreover, notice that, as it was mentioned in [74], the ‘?nitarity index i’ on so(1, 3)i indicates that the Lie ↑ group manifold SO(1, 3) of (local) structure gauge symmetries of the qausets is also subjected to discretization as well. It is reasonable to assume that ?nitary structures have ?nitary symmetries or equivalently and perhaps more popularly, discrete structures possess discrete symmetries. This is in accord with our abiding to a Kleinian conception of (physical) geometry, as noted in footnote 97. On the other hand, we shall see in the next section that the ?nitarity index indicates only that our structures are ‘discrete’ and not that they are essentially dependent on the locally ?nite covering (gauge) Ui of X . In fact, we will see that (from the dynamical perspective) our constructions are inherently gauge Ui -independent and for this reason ‘alocal’ [92, 74, 93]. In other words, the (dynamical) role played by the base localization causet Pi and, in extenso, by the region X of the Lorentzian spacetime manifold that the latter discretizes relative to Ui , is physically insigni?cant. 102 In fact, as noted in both [92] and [93], at the ?nitary poset level one must actually insist on relaxing Hausdor?ness, because a T2 -?nitary substitute in [114] is automatically trivial as a topological space—that is, it carries the discrete topology, or equivalently, it is a completely disconnected set (no arrows between its point vertices). 103 In fact, we could have directly started our ?nsheaf constructions straight from a paracompact and Hausdor? X without coming into con?ict with Sorkin’s results. For instance, already in [75] we applied the entire sheaf-

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2. About the stalk: Lorentzian metric and its orthochronous symmetries. The stalks of the ?i s are occupied by qausets ?i ; in other words, they are the spaces where the (germs of the) continuous local sections of the ?i s take values. These qausets, as it has beeen argued in [74], determine a metric ρi of Lorentzian signature. Thus, as it was emphasized in footnote 17 of 2.2, ρi is not carried by the base space Pi , which is simply a topological space; rather, it concerns directly the (objects living in the stalks of the) relevant ?nsheaves per se. In fact, we may de?ne this metric to be the following ?nsheaf morphism: ρi : Ei↑ ⊕ Ei↑ ?→ Ai (114)

which, like its abstract version ρ in (11), is Ai -bilinear between the (di?erential) Ai-modules ?i concerned and (section-wise) symmetric.104 It follows that the Ai -metric preserving ? ? → ? ? → (local) automorphism group ?nsheaf AutAi Ei↑ |U ∈Pi ≡ Autρi Ei |U ∈Pi is the aforesaid principal ? ? → G -?nsheaf Pi↑ (U ) ≡ Autρi Ei (U ) ≡ SO(1, 3; Ai(U ))↑ i of reticular orthochronous isometries of ↑ the (real) Lorentzian ?nsheaf Ei = (Ei , ρi ) of rank 4.105 Also, in accordance with Sorkin et al.’s remark in [18] that a (locally ?nite) partial order determines not only the topological and the metric structure of the Lorentzian manifold of general relativity, but also its di?erential structure, we witness here that the aforementioned nilpotent Cartan-K¨ ahler (exterior) di?erentials dp i , which as we saw in (111) e?ect vector p p p+1 subsheaf morphisms di : ?i ?→ ?i (Z ? p ≥ 0), derive directly from the algebraic structure of the ?i s—that is to say, again straight from the stalk of the ?nsheaves of qausets without any dependence on the base causet Pi which is simply a causal-topological space. We cannot overemphasize this either: Di?erentiability in our ?nitary scheme, and according to ADG, does not depend on the base space (which is assumed to be simply a topological space); the differential mechanism comes staright from the stalk (ie, from the algebraic objects dwelling in it) and, a fortiori, certainly not from a classical, C ∞ -smooth base spacetime manifold.cohomological panoply of ADG to our ?nsheaves of qausets. 104 In connection with footnote 100, we note that we tacitly assume that Ei↑ = (Ei , ρi ) in (114) is the dual to ?i (ie, ?i = Ei↑? = HomAi (Ei↑ , Ai )). It is also implicitly assumed that ρi in (114) induces a canonical isomorphism

between Ei↑ and its dual ?i analogous to (12). Thus, with a certain abuse of language, but hopefully without causing any confusion, we will assume that ?i ≡ Ei↑ (ie, we identify via ρi ?nitary covectors and vectors) and use them interchangeably in what follows. 105 Since, as noted in footnote 15, speci?c dimensionality arguments do not interest us here as long as the algebras involved in the stalks of our ?nsheaves are (and they are indeed) ?nite dimensional, the reader may feel free to choose an arbitrary, ?nite rank n for our ?nsheaves. Then, the reticular Lorentzian Ai -metric ρi involvedn?1

3. About the physical interpretation. We would like to comment a bit on the physical interpretation of our principal ?nsheaves of qausets and the reticular spin-Lorentzian connections on them. First we must note that Sorkin et al., after the signi?cant change in physical interpretation of the locally ?nite posets involved from topological in [114] to causal in [18, 113, 115, 116, 117] alluded to above, insisted that, while the topological posets can be interpreted as coarse approximations to the continuous spacetime manifold of macroscopic physics, the causets should be regarded as being truly fundamental structures in the sense that the macroscopic Lorentzian manifold of general relativity is an approximation to the deep locally ?nite causal order, not the other way around. Our scheme strikes a certain balance between these two poles. For instance, while we assume a base causet on which we solder our incidence algebras modelling qausets, that causet is also assumed to carry a certain topology—the ‘causal topology’106 —so that it can serve as the background topological space on which to solder our algebraic structures, which in turn enables us to apply ADG to them thus unveil potent di?erential geometric traits of the qausets in the stalks, as described above. This causal topology however, in contradistinction to Sorkin’s T0 -topological posets which model “thickened space-like hypersurfaces” in continuous spacetime [114], is regarded as a theory of ‘thickened’ causal regions in spacetime [86, 74, 93].107 Furthermore, as it has been emphasized in [74], while the non-?at reticular spin-Lorentzian connections Di on the corresponding ?i s can be interpreted as the fundamental operators encoding the curving of quantum causality thus setting the kinematics for ← ? a dynamically variable quantum causality, an inverse system G := {(Pi↑ , Di )} was intuited to ‘converge’ at the operationally ideal (ie, non-pragmatic and ‘classical’ in Bohr’s ‘correspondence principle’ sense [92]) limit of in?nite re?nement or localization of both the base causets and the associated qauset ?bers over them to the classical principal ?ber bundle (P ↑ , D ) of continuous local orthochronous Lorentz symmetries so(1, 3)↑ of the C ∞ -smooth spacetime manifold M of general relativity and the sl(2, C)-valued spin-Lorentzian gravitational connection D on it.108 Since (P ↑ , D ) is the gauge-theoretic version of the kinematical structure of general relativity—the dynamical theory of the classical ?eld of local causality ← ? g?ν ,109 each individual member (Pi↑ , Di) of the inverse system G was interpreted as the kinematics of a locally ?nite, causal and quantal version of (vacuum) Einstein LorentzianSee footnote 96. For more on this, see subsection 4.3 below. ← ? 108 For more technical details about the projective limit of G , the reader must wait until the following subsection. At this point it must be stressed up-front, in connection with footnote 75, that what we actually get at the ← ? projective limit of G is a C∞ -smooth principal bundle (and its spin-Lorentzian connection) over the region X of a ‘generalized di?erential manifold’ (ie, C∞ -smooth) M . 109 For recall that the spacetime metric g?ν (x), for every x ∈ M , delimits a Minkowski lightcone based at x (by the equivalence principle, the curved gravitational spacetime manifold of general relativity is, locally, Minkowski space, ie, ?at, and in this sense general relativity may be viewed as special relativity being localized or ‘gauged’). Thus, the Einstein equations of general relativity, which describe the dynamics of g?ν (which, in turn, can be interpreted as the ?eld of the ten gravitational potentials), e?ectively describe the dynamical changes of (the ?eld of) local causality. All this was analyzed in detail in [74].107 106

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gravity.110 In toto, we have amalgamated aspects from the interpretation of both the ?nitary substitutes and the causets, as follows:111 ‘Coarse causal regions’ are truly fundamental, operationally sound and physically pragmatic, while the classical pointed C ∞ -smooth spacetime manifold ideal.112 Curved ?nsheaves of qausets (Pi↑ ≡ Ei↑, Di ) model the kinematics of dynamical (local) quantum causality in vacuo as the latter is encoded in the f cqv -dynamo Di . A generalized (ie, C∞ -smooth) version of the classical kinematical structure of general relativity, (P ↑ , D ), over the di?erential spacetime manifold M , arises at the ideal and classical (Bohr’s correspondence) limit of in?nite localization of ← ? the qausets—in point of fact, of G .113 4. About ‘reticular’ di?erential geometry. The basic moral of our application of ADG to the ?nitary regime as originally seen in [74] as well as here, but most evidently in [75], is that the fundamental di?erential mechanism which is inherent in the di?erential geometry that we all are familiar with114 is independent of C ∞ -smoothness so that it can be applied in full to our inherently reticular models, or equally surprisingly, to spaces that appear to be ultra-singular and incurably pathological or problematic when viewed from the di?erential manifold’s viewpoint [77, 78, 97]. In our case, what is startling indeed is that none of the usual ‘discrete di?erential mathematics’ (eg, di?erence calculus, ?nite elements or other related Regge calculus-type of methods) is needed in order to address issues of di?erentiability and to develop a full-?edged di?erential geometry in a (locally) ?nite setting. For instance, there appears to be no need for de?ning up-front ‘discrete di?erential manifolds’ and for developing a priori and, admittedly, in a physically rather ad hoc manner a ‘discrete di?erential geometry’ on them115 in order to investigate di?erential geometric properties of ‘?nitary’ (ordered) spaces.116 For they too can be cast under the wider axiomatic, algebraico-sheaf-theoretic prism of ADG as a particular application of the general theory. All in all, it is quite surprising indeed that the basic objects of the usual di?erential geometry like ‘tangent’ vectors (derivations), their dual forms, exteriorAs we shall see in the next section, the actual kinematical con?guration space for the locally ?nite, causal and quantal vacuum Einstein gravity is the moduli space Ai of ?nitary spin-Lorentzian connections Di . As we shall see, projective limit arguments also apply to an inverse system of such reticular moduli spaces. 111 Further distillation and elaboration on these ideas in subsection 4.3. 112 More remarks on ‘coarse causal regions’ will be made in subsection 4.3. 113 This is a concise r? esum? e of a series of papers [92, 86, 74, 93, 75, 90]. Of course, ‘in?nite localization’ requires ‘in?nite microscopic power’ (ie, energy of determination or ‘measurement’ of locution) which is certainly an ideal (ie, operationally non-pragmatic and physically unattainable) requirement. This seems to be in accord with the pragmatic cut-o?s of quantum ?eld theory and the fundamental length LP (the Planck length) that the ‘true’ quantum gravity is expected to posit (and below which it is expected to be valid!), for it is fairly accepted now that one cannot determine the locus of a quantum particle with uncertainty (error) less than LP ≈ 10?35 m without creating a black hole. This seems to be the raison d’? etre of all the so-called ‘discrete’ approaches to quantum spacetime and gravity [74]. 114 Albeit, just from the classical (ie, C ∞ -smooth) perspective. 115 Like for example the perspective adopted in [26, 28, 10, 27]. 116 Like graphs (directed, like our posets here, or undirected), or even ?nite structureless sets.110

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derivatives, Laplacians, volume forms etc, carry through to the locally ?nite scene and none of their discrete (di?erence calculus’) analogues is needed, but this precisely proves the point: One feels, perhaps ‘instinctively’ due to one’s long time familiarity with and the numerous ‘habitual’ (but quite successful!) applications of the usual smooth calculus where the di?erential mechanism comes from the supporting space (ie, it is provided by the algebra C ∞ (M ) of in?nitely di?erentiable functions on the di?erential manifold M ), that in the ‘discrete’ case too some novel kind of ‘discrete di?erential geometry’ must come from a ‘discrete di?erential manifold’-type of base space—as if the di?erential calculus follows from, or at least that it must be tailor-cut to suit, space. In other words, in our basic working philosophy we have been misled by the habitual applications and the numerous successes of the smooth continuum into thinking that di?erentiability comes from, or that it is somehow vitally dependent on, the supporting space. By the present application of ADG to our reticular models we have witnessed how, quite on the contrary, di?erentiability comes from the stalk—ie, from algebras dwelling in the ?bers of the relevant ?nsheaves—and it has nothing to do with the ambient space, which only serves as an auxiliary, and in no way contributing to the said di?erential mechanism, topological space for the sheaf-theoretic localization of those algebraic objects. The usual di?erential geometric concepts, objects and mechanism that relates the latter still apply in our reticular environment and, perhaps more importantly, in spite of it.

Continuous limits of ?nitary simplicial complexes and their associated incidence algebras, regarded as discrete and quantal topological spaces [92, 93], have been studied recently in [130, 131]. In this subsection, always based on ADG, we present the projective limit of the inverse sys← ? tem G = {(Pi↑ , Di )} of principal Lorentzian ?nsheaves of qausets Pi↑ equipped with reticular spin-Lorentzian connections Di which was supposed in [74] to yield the classical kinematical structure of general relativity in its gauge-theoretic guise—that is, the principal orthochronous spin-Lorentzian bundle over the (region X of the) C ∞ -smooth spacetime manifold M of general relativity locally supporting an sl(2, C)-valued (self-dual) smooth connection (ie, gauge potential) 1-form A(+) . We center our study on certain results from a recent categorical account of projective and inductive limits in the category DT of Mallios’ di?erential triads in [81, 82], as well as on results from a treatment of projective systems of principal sheaves (and their associated vector sheaves) endowed with Mallios’ A-connections in [124, 125, 126]. Then, we compare this inverse limit result, at least at a conceptual level and in a way that emphasizes the calculus-free ← ? methods and philosophy of ADG, with the projective limit of a projective family M of compact Hausdor? di?erential manifolds employed in [8] in order to endow the moduli space A/G of gauge-equivalent non-abelian Y-M and gravitational connections with a di?erential geometric

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← ? structure. In fact, we will maintain that an inverse system M of our ?nitary moduli spaces should yield at the projective limit of in?nite localization a generalized version (ie, a C∞ -smooth one) (+) of the classical moduli space A∞ of gauge-equivalent (self-dual) C ∞ -connections on the region X of the smooth spacetime manifold M . The concept-pillar on which ADG stands is that of a di?erential triad T = (A, ?, ?) associated with a K = R, C-algebraized space (X, A). In ADG, di?erential triads specialize to abstract di?erential spaces, while the As in them stand for (structure sheaves of) abstract di?erential algebras of generalized smooth or di?erentiable coordinate functions, and they were originally born essentially out of realizing that the classical di?erential geometry of a manifold X is deduced from its structure sheaf ∞ CX , the latter being for the case at issue the result of the very topological properties117 of the underlying ‘smooth’ manifold X . Thus, in e?ect, the ?rst author originally, and actually quite independently of any previous relevant work, intuited, built and subsequently capitalized on the fact that the algebra sheaf A of generalized arithmetics (or abstract coordinates) is precisely the structure that provides one with all the basic di?erential operators and associated ‘intrinsic di?erential mechanism’ one needs to actually do di?erential geometry—the classical, C ∞ -smooth theory being obtained precisely ∞ when one chooses CM as one’s structure sheaf of coordinates.118 Thus, the objects dwelling in the stalks of A may be perceived as algebras of generalized (or abstract) ‘in?nitely di?erentiable’ (or ‘smooth’) functions, with the di?erential geometric character of the base localization space X left completely undetermined—in fact, it is regarded as being totally irrelevant to ADG.119 In [81], the di?erential triads of ADG were seen to constitute a category DT—the category of di?erential triads. Objects in DT are di?erential triads and morphisms between them represent abstract di?erentiable maps. In DT one is also able to form ?nite products and, unlike the category of smooth manifolds where an arbitrary subset of a (smooth) manifold is not a (smooth) manifold, one can show that every object T in DT has canonical subobjects [81]. More importantly however, in [82] it was shown that DT is complete with respect to taking projective and inductive limits of projective and inductive systems of triads, respectively.120 This is a characteristic di?erence between DT and the category of manifolds where the projective limit of an inversePoincar? e lemma [77, 75]. Yet, we can still note herewith that the ?rst author arrived at the notion of a di?erential triad as a particularization to the basic di?erentials of the classical theory of the amply ascertained throughout the same theory ∞ instrumental role played by the notion of an A(≡ CX )-connection (ie, covariant di?erentiation). 119 ∞ Of course, as also noted earlier in footnote 64, in the classical case (ie, when one identi?es AM ≡ CM ) there is a confusion of the sort ‘who came ?rst the chick or the egg?’, since one tends to identify the underlying space(time) ∞ (ie, the C ∞ -smooth manifold M ) with its structure sheaf CM of smooth functions and, more often than not, one is (mis)led into thinking that di?erentiability—the intrinsic mechanism of di?erential geometry so to speak—comes (uniquely!) from the underlying smooth manifold. This is precisely what ADG highlighted: di?erentiability comes in fact from the structure sheaf, so that if one chooses ‘suitable’ or ‘appropriate’ (to the problem one chooses to address) algebras of ‘generalized smooth’ functions other than C ∞ (M ), one is still able to do di?erential geometry (albeit, of a generalized or abstract sort) in spite of the classical, C ∞ -smooth base manifold. 120 In fact, Papatrianta?llou showed that projective/inductive systems of di?erential triads having either a com118 117

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system of manifolds is not, in general, a manifold.121 Moreover, Vassiliou, by applying ADGtheoretic ideas to principal sheaves (whose associated sheaves are precisely the vector sheaves of ˙ of the triads in the aforesaid ADG) [124, 125, 126], has shown that when the ?at di?erentials ? projective/inductive systems of Papatrianta?llou are promoted (ie, ‘gauged’ or ‘curved’) to A˙ on principal sheaves, the corresponding projective/inductive systems (Pi , D ˙ i )122 connections D have principal sheaves endowed with non-?at connections as inverse/direct limits. Thus, in our locally ?nite case, the triplet Ti = (APi ≡ Ai , DPi ≡ Di , dp i ) is an ADGtheoretic di?erential triad of a (f)initary, (c)ausal and (q)uantal kind. In other words, the category DTf cq having for objects the di?erential triads Ti and for arrows the ?nitary analogues of the triad-morphisms mentioned above is a subcategory of DT called the category of fcq-di?erential?

mon, ?xed base topological space X (write Ti (X )), or a projective/inductive system thereof indexed by the same set of indices (write Ti (Xi )), possess projective/inductive limits. Below, we will see that our projective/inductive ← ? system G = {(Pi↑ , Di )} of ?nitary posets (causets), (principal) ?nsheaves of incidence algebras (qausets) over them and reticular spin-Lorentzian connections on those ?nsheaves, are precisely of the second kind. The reader should also note here that in the mathematics literature, ‘projective’, ‘inverse’ and ‘categorical’ limits are synonymous terms; so are ‘inductive’ and ‘direct’ limits (also known as ‘categorical colimits’). The result from [82] quoted above can be stated as follows: the category DT is complete and cocomplete. This remark, that is to say, that DT is (co)complete, will prove to be of great importance in current research [91] for showing that the category of ?nsheaves of qausets—which is a subcategory of DT—is, in fact, an example of a structure known as a topos [64]—a topos with a non-Boolean (intuitionistic) internal logic tailor-made to suit the ?nitary, causal and quantal vacuum Einstein-Lorentzian gravity developed in the present paper. 121 From a categorical point of view, this fact alone su?ces for regarding the abstract di?erential spaces (of structure sheaves of generalized di?erential algebras of functions and di?erential modules over them) that the ADG-theoretic di?erential triads represent as being more powerful and versatile di?erential geometric objects than C ∞ -manifolds. As also mentioned in [82], it was precisely due to the aforesaid shortcomings of the category of smooth manifolds that led many authors in the past to generalize di?erential manifolds to di?erential spaces in which the manifold structure is e?ectively redundant [108, 109, 80, 51]. In fact, the ?rst author’s di?erential triads generalize both C ∞ -manifolds and di?erential spaces, and, perhaps more importantly for the physical applications, they are general enough to include non-smooth (‘singular’) spaces with the most general, non-functional, structure sheaves [77, 78, 97]. On the other hand, a little bit later we will allude to and, based on ADG and its ?nitary application herein, comment on an example from [8] of an inverse system of di?erential manifolds that yields a di?erential manifold at the projective limit. 122 With (I , ≥) a partially ordered, directed set (net) of indices ‘i’ labelling the elements of the inverse/direct ˙ i ). The systems (Pi , D ˙ i ) are said to be (co)?nal with respect to the index net (I , ≥). We remind system (Pi , D the reader that in our case ‘i’ is the ?nitarity or localization index (ie, locally ?nite open covers Ui of X ? M form a net [114, 87, 74, 75]). 123 The term ‘mixed projective-inductive’ (or equivalently, ‘mixed inverse-direct’) system pertains to the fact ? ← ? that the family T (implicitly) contains both the projective system P = {Pi } of reticular base causets, and the ? → inductive system R of qausets corresponding (by Gel’fand duality) to the aforesaid causets. (Note that we refrain ← ? ? → from putting right-pointing causal arrows over P and R , in order to avoid notational confusion.)

We mentioned earlier the categorical duality between the category P of ?nitary substitutes Pi and poset morphisms between them, and the category Z of the incidence algebras ?i associated with the Pi s and injective algebra homomorphisms between them, which duality is ultimately rooted inWe could have chosen to present the collection {(Pi↑ , Di )} as an inductive family of principal ?nsheaves and their ?nitary connections, since the connections (of any order p) Dp i in each of its terms are e?ectively obtained ← ? ? → by localizing or gauging the reticular di?erentials dp in each term of R . However, that we present G dually, as i an inverse system, is consistent with our previous work [74, 75] and, as we shall see shortly, it yields the same result at the continuum limit (ie, the C∞ -principal bundle). 125 Write ((K) P∞ ,(K) D∞ ) for the C∞ -smooth principal bundle and its non-trivial spin-Lorentzian connection.124

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the general notion of Gel’fand duality.126 In a topological context, the idea to substitute Sorkin’s ?nitary topological posets by incidence Rota algebras was originally aimed at ‘algebraizing space’ [129]—that is to say, at replacing ‘space’ (of which, anyway, we have no physical experience127 ) by suitable (algebraic) objects that may be perceived as living on that ‘space’ and, more importantly, from which objects this ‘space’ may be somehow derived by an appropriate procedure (Gel’fand spatialization). In fact, as brie?y described before, again in a topological context and in the same spirit of Gel’fand duality, the second author substituted Sorkin’s Pi s by ?nsheaves Si of (algebras of) continuous functions that, as we said, are (locally) topologically equivalent (ie, locally homeomorphic) spaces to the Pi s [87]. Here too, the basic idea was, in an operational spirit, to replace ‘space’ by suitable algebraic objects that live on ‘it’, and it was observed that the maximum localization (?nest resolution) of the point-events of the bounded region X of the C 0 -spacetime manifold M by coarse, open regions about them at the inverse limit of a projective 0 system of Pi s, corresponds (by Gel’fand duality) to de?ning the stalks of CX —the sheaf of (germs of) continuous functions on the topological manifold X —at the direct limit of (in?nite localization of) an inductive system of the Si s.128 At the end of [87] it was intuited that if the stalks of the Si s were assumed to be inhabited by incidence algebras which are discrete di?erential manifolds as explained above, at the inverse limit of in?nite re?nement or localization of the projective ← ? system P of Sorkin’s topological posets yielding the continuous base topological space X , the?

corresponding (by Gel’fand duality) inverse-direct system T of ?nitary di?erential triads should ∞ yield the classical structure sheaf AX ≡ (C) CX of germs of sections of (complex-valued)129 smooth C (C) ∞ functions on X and the sheaf ?X of C (X )-bimodules of (complex) di?erential forms, in accordance with Gel’fand duality. There are two issues to be brought up here about this intuition at the end of [87]. First thing to mention is that, as alluded to earlier, it is more accurate to say that, since the incidence algebras are objects categorically or Gel’fand dual to Sorkin’s topological posets, and since the latter form ← ? an inverse or projective system P , the former should be thought of as constituting a direct or ? → inductive system R of algebras possessing K C ∞ (X ) and (K) ?(X ) over it as an inductive limit.130 In fact, as mentioned in the previous paragraph, the stalks of (K) ?X (in fact, of any sheaf! [87]), which are inhabited by germs of sections of C ∞ -smooth (K = R, C-valued) di?erential forms, are obtained precisely at that inductive limit. We may distill all this to the following physical statement which foreshadows our remarks on Gel’fand duality to be presented in the next section: While ‘space(time)’ is maximally (in?nitely) localized (to its points) by an inverse limit of a projective system of Sorkin’s ?nitary posets, the (algebraic) objects that liveSee our more analytical comments on Gel’fand duality in the next section. Again, see more analytical comments on the ‘unphysicality’ of space(time) in the next section. 128 And it should be emphasized that the stalks of a sheaf are the ‘ultra-local’ (ie, maximally localized) point-like elements of the sheaf space [67, 87]. 129 In [92, 86, 87, 93, 74] it was tacitly assumed that we were considering incidence algebras over the ?eld C of complex numbers.127 126

on space(time) (ie, the various physical ?elds) are maximally (in?nitely) localized in the stalks of the ?nsheaves that they constitute by a direct limit of an inductive system of those ?nsheaves. Equivalently stated, ‘space(time)’ is categorically or Gel’fand dual to the physical ?elds that are de?ned on ‘it’. The second thing that should be stressed here, and in connection with footnote 75, is that we ∞ do not actually get the classical di?erential geometric structure sheaf K CX and the corresponding sheaf K ?X of K C ∞ (X )-modules of di?erential forms. In toto, we do not actually recover the clas∞ sical C ∞ -smooth di?erential triad T∞ := (AX ≡ K CX , ?, ?1 X ) at the limit of in?nite localization of the system T , but rather we get the generalized smooth (ie, what we call here C∞ -smooth) triad p p T∞ = (AX ≡ K C∞ X , ?∞ , d∞ ). Of course, by the general theory (ie, ADG), we are guaranteed that? ?

the direct, co?nal system T of ‘generalized discrete di?erential spaces’—that is, the fcq-triads Ti = (Ai , Di , dp i )—yields a well de?ned di?erential structure at the categorical colimit within DT; moreover, according to ADG, it is quite irrelevant whether the di?erential triad at the limit is the classical smooth T∞ of the featureless C ∞ -manifold proper or one for example that is infested by singularities thus most pathological and unmanageable when viewed from the classical C ∞ -manifold perspective [77, 78, 97].131 The point we wish to make here is simply that at the continuum limit we get a, not the familiar C ∞ -smooth, di?erential structure on the continuous topological (C 0 ) spacetime manifold X . This di?erential structure ‘for all practical purposes’ represents for us the classical, albeit ‘generalized’, di?erential manifold, and the direct limiting procedure that recovers it a generalized version of Bohr’s correspondence principle advocated in [92]. That this di?erential structure obtained at the ‘classical limit’ is indeed adequate for accommodating the classical theory will become transparent in the next section where we will see that based on T∞ we can actually write the classical vacuum Einstein equations of general relativity; albeit, in a generalized, ADG-theoretic way independently of the usual C ∞ -manifold. In fact, we will see that these equations are obtained at the inverse limit of a projective system ? ← ? E of vacuum Einstein equations—one for each member of T . 4.2.2 Some comments on real versus complex spacetime and the general use of the number ?elds R and C

As it has been already anticipated in [74, 90], starting from principal ?nsheaves of complex (K = C) incidence algebras carrying non-?at reticular spin-Lorentzian Ai -connections Di as Clinear ?nsheaf morphisms between the ‘discrete’ di?erential Ai -bimodules ?p i (p ≥ 1) in Di , complex (bundles of) smooth coordinate algebras, modules of di?erential forms over them132 and smooth so(1, 3)↑ C -valued connection 1-forms A (over a smooth complex manifold) are expected to emerge at the inductive/projective limit of in?nite re?nement and localization of the qausets and the principal ?nsheaves thereof.133 Thus it may be inferred that in order to recover the realSee footnote 121. That is to say, the generalized ‘classical’, C∞ -smooth di?erential triad T∞ mentioned above. 133 Indeed, in the context of non-perturbative (canonical) quantum gravity using Ashtekar’s new gravitational connection variables, we will see in the next section how a holomorphic Lorentzian spacetime manifold and smooth,132 131

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spacetime continuum of macroscopic relativistic gravity (general relativity), some sort of reality conditions must be imposed after the projective limit, the technical details of which have not been fully investigated yet [130, 131]. The nature of these conditions is a highly non-trivial and subtle issue in current quantum gravity research [14]. On the other hand, starting from incidence algebras over R (K = R), one should be able recover a real C∞ -smooth manifold instead of a complex one at the projective/inductive classical limit’, but then one would not be faithful to the conventional quantum theory with its continuous coherent superpositions over C.134 On the other hand, prima facie it appears to be begging the question to maintain that we have an ‘innately’ or ‘intrinsically ?nitistic’ model for the kinematical structure of Lorentzian quantum spacetime and gravity (and, as we shall contend in the following section, also for the dynamics) when its (noncommutative) algebraic representation employs ab initio the continuum of complex numbers as the ?eld of (probability) amplitudes. For example, in the light of application of ideas from presheaves and topos theory to quantum gravity, Butter?eld and Isham [21], and more recently Isham [55], have also explicitly doubted and criticized the a priori assumption and use of the continuum of either the reals or, a fortiori, of the complexes in quantum theory vis-` a-vis the quest for a genuinely quantum theoresis of spacetime structure and gravity. In [55] in particular, Isham maintains that the use of the arithmetic continua of R (modelling probabilities and the values of physical quantities) and C (probability amplitudes) in standard quantum mechanics is intimately related (in fact, ultimately due) to the a priori assumption of a classical stance against the ‘nature’ of space and time—ie, the assumption of the classical spacetime continuum. In the sequel, in order to make clear-cut remarks on this in relation to ADG, as well as to avoid as much as we can ‘vague dark apostrophes’, by ‘spacetime continuum’ we understand the locally Euclidean arena (ie, the manifold) that (macroscopic) physics uses up-front to model spacetime. Our contention then is that Isham questions the use of R and C in quantum theory precisely because he is motivated by the quest for a genuinely quantum theoresis of spacetime and gravity, for in quantum gravity research it has long been maintained that the classical spacetime continuum (ie, the manifold) must be abandoned in the sub-Planckian regime where quantum gravitational e?ects are expected to be signi?cant.135 Thus, his basic feeling is that the conventional quantum theory, with its continuous superpositions over C and probabilities in R, which it basically inherits from the classical spacetime manifold, must be modi?ed vis-` a-vis quantum gravity. In toto, if the manifold has to go in the quantum deep, so must the number ?elds R and C of the usual quantum mechanics, with a concomitant relatively drastic modi?cation of the usual quantum formalism to suit the non-continuum basecomplex (self-dual) connections on it are the basic dynamical elements of the theory. 134 And indeed, in [86, 93, 75] the C-linear combinations of elements of the incidence algebras where physically interpreted as coherent quantum superpositions of the causal-topological arrow connections between the eventvertices in the corresponding causets. In fact, it is precisely this C-linear structure of the qausets that quali?es them as sound quantum algebraic analogues of causets, which are just associative multiplication structures (arrow semigroups or monoids or even poset categories). Also, in connection with footnote 87, we emphasize that it is the linear structure of qausets (prominently absent from causets) that gives them both their di?erential (geometric) and their quantum character. 135 For instance, see the two opening quotations.

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space(time).136 Perhaps the use from the beginning of one of the ?nite number ?elds Zp 137 for c-numbers would be a more suitable choice for our reticular models, but then again, what kind of quantum theory can one make out of them?138 The contents of this paragraph are captured nicely by the following excerpt from [55]: “...These number systems [ie, R and C] have a variety of relevant mathematical properties, but the one of particular interest here is that they are continua, by which—in the present context—is meant not only that R and C have the appropriate cardinality, but also that they come equipped with the familiar topology and di?erential structure that makes them manifolds of real dimension one and two respectively. My concern is that the use of these numbers may be problematic in the context of a quantum gravity theory whose underlying notion of space and time is di?erent from that of a smooth manifold. The danger is that by imposing a continuum structure in the quantum theory a priori, one may be creating a theoretical system that is fundamentally unsuitable for the incorporation of spatio-temporal concepts of a non-continuum nature: this would be the theoretical-physics analogue of what a philosopher might call a ‘category error’...” while, two years earlier [21], Butter?eld and Isham made even more sweeping remarks about the use of smooth manifolds in physics in general, and their inappropriateness vis-` a-vis quantum gravity: “...the ?rst point to recognise is of course that the whole edi?ce of physics, both classical andquantum, depends upon applying calculus and its higher developments (for example, functional analysis and di?erential geometry) to the values of physical quantities...why should space be modelled using R? More speci?cally, we ask, in the light of [our remarks above about the use of the continuum of the real numbers as the values of physical quantities]: Can any reason be given apart from the (admittedly, immense) ‘instrumental utility’ of doing so, in the physical theories we have so far developed? In short, our answer is No. In particular, we believe there is no good a priori reason why space should be a continuum; similarly, mutatis mutandis for time. But then the crucial question arises of how this possibility of a non-continuum space should be re?ected in our basic theories, in particular in quantum theory itself, which is one of the central ingredients of quantum gravity...”139

At this point it must be emphasized that in ADG, R and C enter the theory through the generalized arithmetics—the structure sheaf AX , which, as noted earlier, is supposed to be a sheaf of commutative K = R, C-algebras (ie, K = R, C ?→ A). In turn, these arithmetics are invoked only when one wishes to represent local measurements and do with them generalSee below. With ‘p’ a prime integer. 138 Chris Isham in private communication. 139 Excerpt from “Whence the Continuum?” in [21]. These remarks clearly pronounce our application here of ADG, which totally evades the usual C ∞ -calculus, to ?nitary Lorentzian quantum gravity (see also remarks below).137 136

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calculations with the vector sheaves E employed by ADG.140 It is at this point that the basic assumption of ADG that the E s involved are locally free A-modules of ?nite rank n–that is to say, locally isomorphic to An —comes in handy, for all our local measurements and calculations involve A, An and, in extenso, the latter’s natural local transformation matrix group AutE (U ) = E ndE (U )? ≡ Mn (A(U ))? . Thus, real and complex numbers enter our theory through ‘the backdoor of measurement and calculation’, in toto, through ‘geometry’ as understood by ADG.141 On the other hand, and in connection with the last footnote, since the constructions of ADG are genuinely independent of (the usual calculus on) C ∞ -manifolds,142 , whether real (analytic) or complex (holomorphic), [67, 68, 77, 78, 74, 70, 75, 73], Isham’s remarks that the appearance of the arithmentic continua in quantum theory are due to the a priori assumption of a classical spacetime continuum—a locally Euclidean manifold—do not a?ect ADG. Of course, we would actually like to have at our disposal the usual number ?elds in order to be able to carry out numerical calculations (and arithmetize our abstract algebraic sheaf theory) especially in the (quantum) physical applications of ADG that we have in mind.143 We may distill all this to the following: In the general ADG theory, and in its particular ?nitary application to quantum gravity here, the commutative number ?elds, which happen to be locally Euclidean continua (ie, the manifolds R ? R1 and C ? R2 being equipped with the usual differential geometric—ie, C ∞ -smooth—structure), do not appear in the theory from assuming up-front a background spacetime manifold.144 Rather, they are only builtSee sections 2 and 3, and in particular the discussion in 4.3 next. This is in accord with our view of A mentioned earlier as the structure carrying information about the ‘geometry’, about our own measurements of ‘it all’ (see footnotes 17, 41, the end of 2.3 and subsection 4.3 next). In agreement with Isham’s remarks in [55] brie?y mention